Mathematics High School

## Answers

**Answer 1**

0.123412341234... is **equal** to the fraction 1234/9999.

(a) To convert 0.7777... to a fraction, let x = 0.7777....

Then, multiplying both sides by 10, we get:

10x = 7.7777...

Subtracting the left-hand sides and the right-hand sides of these two equations, we get:

10x - x = 7.7777... - 0.7777...

Simplifying, we get:

9x = 7

**Dividing** both sides by 9, we get:

x = 7/9

Therefore, 0.7777... is equal to the fraction 7/9.

(b) To convert -99.9999... to a fraction, let x = -99.9999....

Then, multiplying both sides by -1, we get:

-x = 99.9999...

Dividing both sides by 100, we get:

-x/100 = 0.999999...

Since the decimal 0.999999... is equal to 1, we can rewrite the equation as:

-x/100 = 1

Multiplying both sides by -100, we get:

x = -100

Therefore, -99.9999... is equal to the fraction -100.

(c) To convert 0.09090909... to a fraction, let x = 0.09090909....

Then, **multiplying** both sides by 100, we get:

100x = 9.090909...

Subtracting the left-hand sides and the right-hand sides of these two **equations,** we get:

100x - x = 9.090909... - 0.090909...

Simplifying, we get:

99x = 9

Dividing both sides by 99, we get:

x = 1/11

Therefore, 0.09090909... is equal to the fraction 1/11.

(d) To convert 0.123412341234... to a fraction, let x = 0.123412341234....

Then, multiplying both sides by 10000, we get:

10000x = 1234.12341234...

**Subtracting** the left-hand sides and the right-hand sides of these two equations, we get:

10000x - x = 1234.12341234... - 0.12341234...

Simplifying, we get:

9999x = 1234

Dividing both sides by 9999, we get:

x = 1234/9999

Therefore, 0.123412341234... is equal to the **fraction **1234/9999.

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## Related Questions

many quantitative models of decision theory are based on comparing a single measure of effectiveness, generally some form of utility to the decision maker.a. true b. false

### Answers

The statement many quantitative models of** decision making **are based on comparing a single measure of effectiveness, generally some form of utility to the decision maker is true. So, the correct optio is option a. true.

Many quantitative models of decision theory, such as **Expected Utility Theory **and also **Multi-Attribute Utility Theory**, involve comparing a single measure of effectiveness, typically some form of utility, to the decision maker. This allows for the **evaluation **and comparison of different options based on their expected outcomes and potential risks.

It helps the decision maker determine the best course of action.

Decision theory is a branch of probability. It is concerned with** analytic philosophy**. It deals with the decision making processes based on **probabilities **depending on the various factors and the numerical consequences of the outcome.

Therefore, the statement about the **decision theory** based on comparing a single measure of effectiveness is true.

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24. A store has only 2 white shirts and 10 black shirts. If only white shirts are to be added to the inventory so that the probability of selecting a white shirt from all the shirts is 3/5 , how many white shirts need to be added?

### Answers

The total number of **white shirts **needed to be added by the **store **would be 13.

Word problem

Assuming that x is the number of** white shirts** that need to be added. After adding these shirts, the total number of shirts in the store will be:

2 + x white shirts and 10 black shirtstotal shirts = 12 + x shirts.

If the **probability **of selecting a white shirt from all the shirts is 3/5, then we can set up the following equation and solve for x accordingly:

(2 + x) / (12 + x) = 3/55(2 + x) = 3(12 + x)10 + 5x = 36 + 3x2x = 26x = 13

Therefore, the **store **needs to add 13 **white shirts. **

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Use arrow notation to describe the translation.

W(-2,-6) to W'(4,9)

### Answers

The **arrow notation** that describes the **translation** is (x, y) ⇒ (x + 3, y + 6)

Using the arrow notation to describe the translation.

From the question, we have the following parameters that can be used in our computation:

W(-2,-6) to W'(4,9)

The translation is the **difference **between the **pairs**

So, we have

(x, y) = (9 - 6, 4 + 2)

Evaluate

(x, y) = (3, 6)

Hence, the **translation **is (x, y) ⇒ (x + 3, y + 6)

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Find the electric field in the three regions: (i) r < a, (ii) a < r < b, (iii) r > b. Plot |e| as a function of r, for the case b

### Answers

The boundaries between the various plot sections of **electric field **will be two vertical lines.

In this work, the **distance **from a point charge must be calculated under three different circumstances: (1) when the distance is less than the radius of the point charge; (2) when the distance is greater than the **radius **of the outer sphere; and (3) when the distance is between the radii of two concentric charged spheres.

Numerous calculations that connect the **charge **distribution and distance to the electric field are used to determine the electric field at each site. The absolute value of the electric field is then displayed as a function of distance using the numerous formulas that were employed, and the different places where the electric field varies are identified.

The boundaries between the various plot sections will be two vertical lines.

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I am lost..please help don't know where I need to find first!

### Answers

The value of the **angle x** in the given polygon image is: x = 54°

How to find the angle in the polygon?

The formula to find the **interior angles** of a **polygon** is:

θ = 180(n - 2)/n

Thus, for a **pentagon** with 5 sides, we have:

θ = 180(5 - 2)/5

θ = 540/5

θ = 108°

**Interior angle** of a square is 90 degrees and since angle at a point is 360 degrees then we have:

x + 108 + 108 + 90 = 360

x = 360 - 306

x = 54°

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The sides of a triangular field are in the ratio of 8:9:10 and the perimeter is 540 m then, a) Write down the formula of finding the area of a triangle having all sides are not equall by Find the area of the land. c)Find the total cost for the fencing the land three times at the rate of Rs. 25 per meter of the wire.

### Answers

a) Area = √[s(s-a)(s-b)(s-c)] b) the area of the land 225.83 m^2 c) the** total cost** for the fencing the land three times at the rate of Rs. 25 per meter of the wire. is Rs. 43,500

How to determine the formula of finding the area of a triangle having all sides are not equall

a) The formula to find the area of a triangle when all sides are not equal is Heron's formula:

**Area** = √[s(s-a)(s-b)(s-c)]

where s is the semi-perimeter of the triangle, and a, b, and c are the lengths of its sides.

To find the semi-perimeter, we add up the three sides and divide by 2:

semi-perimeter = (8+9+10)/2 = 27/2

Now we can substitute into the formula:

b) **Area** = √[27/2(27/2-8)(27/2-9)(27/2-10)]

= √[27/2(11/2)(9/2)(7/2)]

= √(27^2/4 * 11 * 9 * 7) / 4

= √51030.1875

≈ 225.83 m^2

b) To fence the land three times, we need to find the perimeter of the triangle and multiply by 3:

**Perimeter** = 8x + 9x + 10x = 27x

We know that the perimeter is 540 m, so we can solve for x:

27x = 540

x = 20

Now we can find the actual lengths of the sides:

8x = 8(20) = 160

9x = 9(20) = 180

10x = 10(20) = 200

The** total length **of wire needed to fence the land three times is the perimeter multiplied by 3:

Total length of wire = 3(160 + 180 + 200) = 1740 m

The cost of the wire is Rs. 25 per meter, so the total cost is:

**Total cost** = 1740 * 25 = Rs. 43,500

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A blimp provides aerial television views of a soccer game. The television camera sights the stadium at a 6° angle of depression. The altitude of the

blimpis 400m. What is the line-of-sight distance from the television camera to the base of the stadium? Round to the nearest hundred meters.

### Answers

The line-of-sight distance from the television camera to the case of the stadium is [tex]\text{x = 3826.71 m}[/tex]

Given data

From Δ ABC

AB = The altitude of the blimp is = 400 m

AC = The line-of-sight distance from the television camera to the case of the stadium = x

Now

[tex]\text{sin} \ 6=\dfrac{\text{AB}}{\text{AC}}[/tex]

[tex]x=\dfrac{\text{AB}}{\text{sin} \ 6}[/tex]

[tex]x=\dfrac{400}{\text{sin} \ 6}[/tex]

[tex]\bold{x = 3826.71 \ m}[/tex]

**Therefore the line-of-sight distance from the television camera to the case of the stadium.**

A tree is 55 meters tall. What is its height in feet? Use the following conversion: 1 meter is 3.3 feet.

### Answers

So you're telling me, you're seriously in high school and **do not know this.**

Ah, whatever.

Tree height = 55 meters

Tree height in feet = ?

**1 m = 3.3 feet**

**Tree height in feet = 55 x 3.3 = 181.5 feet.**

The required answer is the height of the tree is provided in meters (55 meters) in the given problem and **convert **it to 181.5 feet. By **multiplying **the given height by the conversion factor of 3.3 feet/meter, we obtain the height of the tree in feet

Feet to meter **conversion**:

To convert a value from feet to meters, we use the conversion factor of 1 meter equals 3.3 feet. This means that for every 1 meter, there are 3.3 feet. To **convert **a measurement from feet to meters, we multiply the given value in feet by the conversion factor.

To **convert **the height of a tree from meters to feet, we can use the conversion factor: 1 meter is equal to 3.3 feet. Let's solve the problem step by step:

Step 1: Multiply the given **height **in meters (55) by the conversion factor (3.3).

55 meters x 3.3 feet/meter

Step 2: Perform the **multiplication **operation:

55 x 3.3 = 181.5 feet

Step 3: Write the result as the height in feet:

The **height **of the tree is 181.5 feet.

Hence, the **height **of the tree is provided in meters (55 meters) in the given problem , and to **convert **it to 181.5 feet. By multiplying the given height by the **conversion factor** of 3.3 feet/meter, we obtain the height of the tree in feet.

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is there a statistically significant difference in weight after the intervention program when comparing across racial groups? please state the p-value from spss and write a final take-home interpretation using a significance level of 0.05

### Answers

If the** p-value** is greater than 0.05, then there is evidence to conclude that there is a **significant difference**.

The p-value is a measure of the strength of evidence against the null hypothesis, which is the **hypothesis **that there is no difference between groups. A small p-value indicates that there is strong evidence against the null hypothesis, and that the observed difference is likely not due to chance.

If the p-value is less than 0.05, which is a commonly used significance level in many fields, it means that there is sufficient **evidence **to reject the null hypothesis and conclude that there is a statistically significant difference between groups.

Now, to answer your question, it would be helpful to know the exact p-value from the **SPSS analysis.** If the p-value is less than 0.05, then there is evidence to suggest that there is a significant difference in weight after the intervention program across racial groups.

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Life insurance companies take risks much like arcade game owners take risks. Mollie has a booth on a popular beach boardwalk. She charges $2 per game. Winners receive a $5 prize. The probability of winning the game is 0.1.

### Answers

** Mollie **can expect to lose **$1.50 **per game played.

To solve this problem

What is **Mollie's **expected profit per game, assuming players play only once?

First, let's calculate the **expected revenue **per game:

Probability of winning: 0.1

Revenue from winners: $5

Probability of losing: 0.9

Revenue from losers: $0

Expected revenue per game = (0.1 x $5) + (0.9 x $0) = $0.50

Now, let's calculate the expected cost per game:

Cost per game: $2

Expected cost per game = $2

Therefore, Mollie's expected profit per game is:

Expected profit per game = Expected revenue per game - Expected cost per game

Expected profit per game = $0.50 - $2

Expected profit per game = -$1.50

This means that, on average, Mollie can expect to lose **$1.50** per game played.

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In Problems 9–16 decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equation. Do not solve the equation. 9. 3y" + 2y' + 8y = 12 + 4t - té' sint 10. y" – y' + y = (e' ++)2 11. y" – 6y' – 4y = 4 sin 3t – e312 +1/t 12. y" + y' + ty = e' + 7 13. y" – 2y' + 3y = cosht + sin’t 14. 2y" + 3y' – 4y = 2t + sin’t + 3 15. y" + e'y' + y = 7 + 3t 16. 2y" – y' + 6y = te sint - 8t cos 3t + 10'

### Answers

The method of **undetermined coefficients** together with superposition can be applied and cannot be applied to: Can be applied: 9, 10, 11, 12, 13, 14, 15, Cannot be applied: 16

The method of undetermined coefficients together with **superposition** is a technique used to find a particular solution to a non-homogeneous linear differential equation. The method involves assuming a particular form for the solution and then finding the coefficients that satisfy the equation.

The **superposition principle** states that the sum of two or more particular solutions is also a particular solution, so once the coefficients are found for each particular solution, the general solution can be obtained by adding them together.

The method of undetermined coefficients is used when the non-homogeneous term of the differential equation is of a certain form, such as a polynomial, exponential, or **trigonometric function**. The form of the particular solution is then chosen based on the form of the non-homogeneous term.

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In the last soccer game, Brandon scored 2 points more than one third of his team's total points. Let n represent the number of points Brandon's team scored. Write an expression for the number of points Brandon scored.

### Answers

Let's call the number of points Brandon's team scored "n".

According to the problem, Brandon scored "2 points more than one third of his team's total points".

One third of the team's total points is (1/3)n.

So, Brandon scored (1/3)n + 2 points.

Therefore, the expression for the number of points Brandon scored is (1/3)n + 2.

Problem 1. State, with a brief explanation, whether the following propositions are true or false. (a) (5 pts) Every homogeneous linear system has a solution. (b) (5 pts) Let A, B, and C be three matrices of the appropriate sizes. Then, the following holds. AB = AC and A 0 =B=0 (c) (5 pts) If A #0 and B + are square matrices of the same order, then AB #O. (

d) (5 pts) If A is skew symmetric, then AP is skew symmetric for any positive integer p.

### Answers

a) The given proposition "Every **homogeneous linear system** has a solution" is true.

A homogeneous linear system is in the form Ax = 0, where A is a matrix and x is a vector of unknowns. This system always has at least the trivial solution (x = 0), where all unknowns are set to zero.

b) The given proposition "Let A, B, and C be three **matrices **of the appropriate sizes. Then, the following holds: AB = AC and A ≠ 0 imply B = C" is true.

If AB = AC and A ≠ 0, we can left-multiply both sides by the inverse of A (A×(-1)). Then, A×(-1)(AB) = A×(-1)(AC) which simplifies to B = C.

c) The given proposition If A ≠ 0 and B ≠ 0 are square matrices of the **same orde**r, then AB ≠ 0 is False.

Even if A and B are nonzero square matrices, their product AB could still be a zero matrix. A counterexample is A = [[1, 1], [1, 1]] and B = [[1, -1], [-1, 1]].

d) The given proposition "If A is skew-symmetric, then A×p is skew-symmetric for any positive integer p" is true.

A skew-symmetric matrix A satisfies the property that A×T = -A. By repeatedly multiplying a skew-symmetric matrix A by itself, the resulting matrix A×p will also be skew-symmetric for any positive integer p. This can be proven by induction.

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let w be the union of the first and third quadrants in the xyplane. that is, let W = [X,Y] :xy ≥0

(a) Is 0 = [0 0 ] in W?

(b) If u is in W and c is any scalar, is cu in W? Why?

(c) Find specific vectors u and v such that u + v is not in W.

(d) Is W a subspace of R2?

### Answers

W fails to satisfy the closure property of a **subspace**, and is not a subspace of R2.

(a) Yes, 0 = [0 0] is in W because 0*0 = 0 and 0 is non-negative.

(b) Yes, if u is in W and c is any scalar, then cu is also in W. This is because if xy ≥ 0 and c ≥ 0, then (cu)x = c(ux) ≥ 0 since ux ≥ 0. Similarly, if xy ≥ 0 and c ≤ 0, then (cu)x = c(ux) ≤ 0 since ux ≥ 0. Therefore, cu is also in W.

(c) One possible choice for u and v such that u + v is not in W is u = [1 0] and v = [0 -1]. Then u and v are both in the first quadrant, but their sum u + v = [1 -1] is in the second** quadrant** and thus not in W

(d) No, W is not a subspace of R2 because it is not closed under vector addition.** Specifically,** if u and v are in W, then their sum u + v may not be in W, as shown in part (c). Therefore, W fails to satisfy the closure **property** of a subspace, and is not a subspace of R2.

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c) Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

f(x) = 7x − 1

f(1/2) =

f(2.5) =

f(-1)=

f(1/4)=

D) Use a calculator to evaluate the function at the indicated values. Round your answers to three decimals.

g(x) = (1/3)^x+1

g(1/2)=

g( √ 3)=

g(-3.5)=

g(-1.4)=

E) A bacteria culture contains 1500 bacteria initially and doubles every hour.

1.Find a function N that models the number of bacteria after t hours.

N(t)= __________

2. Find the number of bacteria after 24 hours.

### Answers

Therefore, there are **approximately** 402,653,184,000 bacteria after 24 hours.

Can we evaluate it at the given values using the given function f(x)=7x-1

c) Using the given function f(x) = 7x - 1, we can **evaluate** it at the given values as follows:

f(1/2) = 7(1/2) - 1 = 2.5 - 1 = 1.5

f(2.5) = 7(2.5) - 1 = 17.5 - 1 = 16.5

f(-1) = 7(-1) - 1 = -7 - 1 = -8

f(1/4) = 7(1/4) - 1 = 1.75 - 1 = 0.75

Therefore, f(1/2) = 1.5, f(2.5) = 16.5, f(-1) = -8, and f(1/4) = 0.75.

d) Using the given function g(x) = (1/3)^(x+1), we can evaluate it at the given values as follows:

g(1/2) = (1/3)^(1/2+1) = (1/3)^(3/2) ≈ 0.192

g(√3) = (1/3)^(√3+1) ≈ 0.011

g(-3.5) = (1/3)^(-3.5+1) = (1/3)^(-2.5) ≈ 3.755

g(-1.4) = (1/3)^(-1.4+1) = (1/3)^(-0.4) ≈ 1.427

Therefore, g(1/2) ≈ 0.192, g(√3) ≈ 0.011, g(-3.5) ≈ 3.755, and g(-1.4) ≈ 1.427.

e) The function that models the number of **bacteria **after t hours is given by:

N(t) = 1500 * 2^(t/1)

where 1500 is the initial number of bacteria and 2^(t/1) represents the number of times the bacteria double in t hours.

To find the number of bacteria after 24 hours, we can **substitute **t = 24 into the above function and evaluate:

N(24) = 1500 * 2^(24/1) = 1500 * 2^24 ≈ 402653184000

Therefore, there are approximately 402,653,184,000 bacteria after 24 hours.

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Write the equation in standard form for the circle with center (3,10) and radius 2.

### Answers

The **equation** of the circle is [tex](x - 3)^2 + (y - 10)^2 = 4[/tex]

What is circle?

A circle is a two-dimensional geometric shape that is defined as the set of all points that are equidistant from a fixed point, called the center. The distance from the center to any point on the circle is called the radius.

Circles have many interesting properties and are commonly studied in geometry. Some of the key properties of circles include:

All points on the circumference of a circle are equidistant from the center.

The diameter of a circle is a line that passes through the center and touches two points on the circumference. The diameter is twice the length of the radius.

The circumference of a circle is the distance around the outside edge. It is equal to the diameter multiplied by pi (π), which is a mathematical constant that is approximately equal to 3.14.

The area of a circle is the amount of space contained within the circumference. It is equal to pi multiplied by the square of the radius.

Now The standard form of the equation of a **circle **is:

[tex](x - h)^2 + (y - k)^2 = r^2[/tex]

where (h, k) is the **center **of the circle and r is the radius.

In this case, the center is (3,10) and the radius is 2. So we can plug these values into the standard form **equation **and simplify:

[tex](x - 3)^2 + (y - 10)^2 = 2^2[/tex]

[tex](x - 3)^2 + (y - 10)^2 = 4[/tex]

Therefore, the equation of the circle with center (3,10) and radius 2 is [tex](x - 3)^2 + (y - 10)^2 = 4[/tex] in **standard **form.

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The diagram shows Kai's garden.

It is in the shape of a right-angled triangle.

20m

15m

Not to scale

Kai is going to spread grass seed on the garden.

• A bag of grass seed covers an area of 35 m².

Each bag of grass seed costs £8.99.

Kai can only buy whole bags of grass seed.

Kai buys the least number of bags needed for the garden.

Calculate the cost of buying the bags of grass seed that Kai needs.

You must show your working.

### Answers

The area of Kai's garden is **150 sq m**. The number of bags of grass seed needed is **5** and the cost of grass seed is **£49.95**

**Area** of Kai's triangle garden = [tex]\frac{1}{2} *b*h[/tex]

where b is the base of the triangle

and h is the height of the triangle

b = **20 m**

h = **15 m**

Area of triangle garden = [tex]\frac{1}{2}*20*15[/tex]

= **150 sq m**

One bag of grass seeds cover = **35 sq m**

150 sq m requires =** 150 / 35** bags of grass seed

= **4.28 bag** is required

Since Kai needs to buy a whole bag, **5 bags** of grass seed is bought

Cost of 1 bag of grass seed = **£8.99**

Cost of 5 bags of grass seed = 8.99 * 5 = **£49.95**

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Evaluate this expression (measures are in radians)

sin 3π/4 x cos 5π/3

### Answers

**Answer:**

=√2/4

**Step-by-step explanation:**

[tex]sin(3\pi /4) = \frac{\sqrt{2}}{2} \\\\and \\\\cos(5\pi /3) = \frac{1}{2} \\\\[/tex]

Therefore, sin(3π/4) × cos(5π/3) = √2/2 × 1/2 = √2/4

A jug of lemon slush is made up of 3/5 litre of water and 1/2 litre of lemon juice how much drink in the jug altogther? Express ur answer in form of a mixed number provide full solution

### Answers

Total **amount **of drink in the jug is 1 1/10 litre.

How much drink in the jug altogther?

To find the total amount of **drink **in the jug, we need to add the amount of water and the amount of lemon juice:

Amount of water = 3/5 litre

Amount of lemon juice = 1/2 litre

We need to find a common denominator to add these **fractions**. The smallest common multiple of 5 and 2 is 10. So we can rewrite the fractions with denominators of 10:

Amount of water = (3/5) x (2/2) = 6/10 litre

Amount of lemon juice = (1/2) x (5/5) = 5/10 litre

Now we can add the fractions:

Total amount of drink = 6/10 + 5/10 = 11/10 litre

Since the numerator is greater than the denominator, we have an improper fraction. We can convert it to a mixed number by dividing the **numerator **by the denominator:

11 ÷ 10 = 1 with a remainder of 1

So the total amount of drink in the jug is 1 1/10 litre.

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Find the trigonometric ratio

### Answers

**Answer: 4/5**

**Step-by-step explanation:**

Sin ∅ = [tex]\frac{opposite}{hypotenuse}[/tex]

- opposite means, the length of the side opposite of the angle

-trig can only be sued in a right triangle, so the hypotenuse is always the side opposite of the right angle, or the longest side

Your question

sin A= [tex]\frac{32}{40}[/tex] you can reduce this ratio by 8, meaning you can divide the top and bottom by 8 to get a reduced fraction

sin A = [tex]\frac{4}{5}[/tex]

make sure to write an explanation!!!

### Answers

**Answer: 24**

**Step-by-step explanation:**

First, you need to find the slope length a single side of the square.

d=√((x2 – x1)² + (y2 – y1)²)

d=√((6 – 1)² + (1 – 2)²)

d=√(5² -1²)

=√(25 -1)

=√(24)

=√(24)

=4.8989794855663561963945681494118

Secondly, you need to use the area formula. side * side

4.8989794855663561963945681494118*4.8989794855663561963945681494118= 24

So, your answer is 24.

Hope this helps! :)

A bag of M&Ms is dumped on the table and you count 97 blue pieces. Estimate how many total M&Ms were in the bag. 4) According to Mars, the makers of M&Ms, their factory produces 25% blue M&Ms, 25% orange, and 12.5% for each of the remaining colors. How do the tables numbers compare to these? Calculate the difference between these values and our class averages for each color. 5) What are some reasons why the M&M bags would differ from the expected distribution?

### Answers

We can **estimate** that there were approximately 388 M&Ms in the bag. 5. The M&M bags may not be **distributed** as planned for a number of reasons.

What is percentage?

When a **number** is expressed as a fraction of 100, the denominator is always fixed at 100. It is frequently used to convey ratios, **proportions**, or rates in a way that is easier to grasp and compare.

On a test, for instance, if a student correctly answers 20 of the 25 questions, we can calculate their percentage by dividing 20 by 25, then multiplying the result by 100:

20/25 x 100 = 80%

Given that, you count 97 blue pieces, and they are 25%.

Thus,

0.25x = 97

x = 388

Hence, we can estimate that there were approximately 388 M&Ms in the bag.

5. The M&M bags may not be distributed as planned for a number of reasons. There is a chance that the sample of M&Ms we saw is not entirely representative of the population of M&Ms in the bag due to random sampling error. Another explanation is that there may be some natural fluctuation in the ratios of various colours in each batch of M&Ms because the manufacturing procedure at the factory is not always exact.

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Recall that the constant term, C, cancels out when evaluating a definite integral and therefore, the C is traditionally omitted in the calculations. Evaluate the definite integral. integral_1/5^50/x^3 dx = [-25x^-2]+1/5^1 = F()-F (1/5) =

### Answers

That the constant of integration C **cancels** out when we evaluate the definite integral, so we do not need to include it in the final answer.

We can evaluate the definite **integral** using the power rule of integration, which states that

∫ [tex]x^n dx = (x^(n+1))/(n+1) + C[/tex]

where C is the constant of integration.

So, for the given definite integral, we have:

∫ [tex](1/(5^(50)x^3)) dx[/tex]

[tex]= (-1/2) * (1/(5^(50))) * (1/x^2) + C[/tex]

**Evaluating** this from x = 1 to x = 1/5, we get:

[tex]= (-1/2) * (1/(5^(50))) * (1/(1/5)^2) - (-1/2) * (1/(5^(50))) * (1/1^2)[/tex]

[tex]= (-1/2) * (5^2) * (5^(50)) + (1/2) * (5^(50))[/tex]

[tex]= - (1/2) * (5^(52)) + (1/2) * (5^(50))[/tex]

[tex]= (1/2) * (5^(50)) * (1 - 5^2)[/tex]

[tex]= (1/2) * (5^(50)) * (-24)[/tex]

[tex]= -12 * (5^(50))[/tex]

So, the value of the **definite **integral is [tex]-12 * (5^(50))[/tex]. Note that the **constant** of integration C cancels out when we evaluate the definite integral, so we do not **need **to include it in the final answer.

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Carlos recently saw an ad online for a new version of his tablet. The projected price is $450, and the tablet will be out on the market in about one year. Jose wants to purchase a new tablet but is wondering if he should wait a year. With a 8.75% inflation, what amount would he pay to purchase a tablet today that is the same value as the one he saw online?

A. $410.63

B. $39.38

C. $489.38

D. $453.94

### Answers

**AnswER**

**410 A**

**Step-by-step explanation:**

Suppose we want to test the effect of three new types of flour on the shelf life of cookies made in a certain bakery. The new flours will be tested on six different recipes for chocolate chip cookies.

Complete the sentence about random assignment.

In this experiment, the experimental units are the

. The bakery testing the flours should create

treatment groups, one group for each type of

.

### Answers

In this experiment, the experimental units are the **cookies **made from each of the six different recipes. The bakery testing the flours should create three **treatment **groups, one group for each type of flour, and use random assignment to assign each recipe to one of the three treatment groups.

To do this, the bakery could randomly assign a number (1, 2, or 3) to each recipe, and then assign the recipe to the corresponding treatment group (e.g., recipe 1 to **treatment group **1, recipe 2 to **treatment group **2, etc.). Alternatively, the bakery could use a randomization table or a computer program to randomly assign the recipes to the treatment groups.

After the **cookies **have been baked using the different types of flour, the bakery can measure the shelf life of the **cookies **and compare the results between the **treatment groups**. By using random assignment, the bakery can be more confident that any differences in shelf life are due to the type of flour used and not to other factors that could affect the results.

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what is not an important issue affecting our ability to estimate the properties of a population when using only sample data?

### Answers

that is not important when estimating population properties using** sample data**, as even minor issues can significantly impact the accuracy of estimates.

One important issue affecting our ability to estimate the properties of a population when using only **sample data **is the representativeness of the sample. If the sample is not a good representation of the population, then the estimates based on the sample data may not accurately reflect the true properties of the population. Other factors that can affect estimates'** accuracy** include the sample size and the sampling method used. However, it is difficult to identify issues that are not important when estimating population properties using sample data, as even minor issues can **significantly impact** the accuracy of estimates.

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find the length of the curve y = 4(x − 1)3/2 , 1 ≤ x ≤ 4

### Answers

The** length **of the curve y = 4(x − 1)³/² from x=1 to x=4 is approximately 18.42 units.

This can be found using the formula for **arc length**: ∫(1 to 4)√(1 + (dy/dx)²) dx. After taking the** derivative** of y, we get dy/dx = 6(x-1)½. Plugging this into the formula and simplifying gives us ∫(1 to 4)6(x-1)dx = 18(3/2) = 27. Therefore, the length of the curve is approximately 18.42 units.

To find the length of a curve, we use the formula for arc length which involves taking the **integral **of the square root of 1 + (dy/dx)². In this case, we first take the derivative of y to get dy/dx = 6(x-1)½.

We then plug this into the formula for arc length and simplify the integral. After integrating from x=1 to x=4, we get the length of the curve to be approximately 18.42 units.

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what is the main connection between logarithmic functions and real-world problems

### Answers

Much of the power of logarithms is their usefulness in solving** exponential equations**. Some examples of this include** sound **(decibel measures), earthquakes , the **brightness of stars,** and chemistry (pH balance, a measure of acidity and alkalinity).

**What is exponential equations?**

An exponential equation is an** equation **in which a** variable **occurs in the exponent. For example, y = 5x is an **exponential **equation since the exponent is the variable x (also said as "5 to the power of x"), while y = x5 is not an exponential equation since the exponent is 5 and not a variable.

**What is logarithmic functions?**

The basic logarithmic **function** is of the form f(x) = logax (r) y = logax, where a > 0. It is the inverse of the exponential function ay = x. Log functions include natural logarithm (ln) or common** logarithm** (log).

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A house was valued at $275,000. Over several years, the value decreased by 18%, giving the house a new value.

(a) Fill in the blank to write the new value in terms of the old value. Write your answer as a decimal.

New value = __ x Old value

(b) Use your answer in part (a) to determine the new value.

New value: __

### Answers

the new value of the **house** is $225,500. we can solve this by **substituting** the given information into the **equation** from part (a)

**what is equation ?**

An equation is a** mathematica**l statement that uses an equal sign to show that two expressions have the same value. Equations are used to represent a **wide range** of relationships between variables

In the given question,

(a) To write the new value in terms of the old value, we can use the fact that the value decreased by 18%, which means that the new value is equal to 100% - 18% = 82% of the old value. So, we can write:

New value = 0.82 x Old value

(b) To determine the new value, we can substitute the given information into the equation from part (a):

New value = 0.82 x 275,000

New value = 225,500

Therefore, the new value of the house is $225,500.

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we would like to conduct a hypothesis testing to see if mean vehicle speed at a particular off-ramp on a highway segment is 45 mph. we have collected vehicle speed data for 60 vehicles to create a sample. the mean speed for these 60 observations is 46.5 mph. assuming the sample standard deviation is 4.1, conduct a hypothesis testing to see if population mean speed on this off-ramp is 45 mph at a 95% confidence level. what is ? group of answer choices

### Answers

Answer: The test statistic ? is 3.17.

**Step-by-step explanation:**

To conduct a hypothesis testing to see if the population mean speed on the off-ramp is 45 mph at a 95% confidence level, we need to set up the null and alternative hypotheses:

Null hypothesis: The population mean speed on the off-ramp is 45 mph (μ = 45)

Alternative hypothesis: The population mean speed on the off-ramp is not 45 mph (μ ≠ 45)

We can use a two-tailed t-test since we do not have any prior knowledge about the direction of the difference.

Next, we need to calculate the test statistic:

t = (sample mean - hypothesized population mean) / (sample standard deviation / sqrt(sample size))

t = (46.5 - 45) / (4.1 / sqrt(60))

t = 3.17

The degrees of freedom (df) for this test is (n - 1) = 59. We can find the critical value using a t-table or a t-distribution calculator for a 95% confidence level with df = 59:

t_critical = ±2.002

Since our calculated t-value (3.17) is outside the range of the critical values (-2.002, 2.002), we can reject the null hypothesis and conclude that there is sufficient evidence to suggest that the population mean speed on the off-ramp is not 45 mph at a 95% confidence level.

Answer: The test statistic ? is 3.17.