The teachers at Eastside High School were asked the number of years they have been teaching. The results are as follows.

3, 4, 9, 12, 7, 5, 4, 12, 3, 9, 11, 4, 14, 9, 2, 10, 3, 10, 9, 7

Use the data to fill in the box-and-whisker plot below.

The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.

In simple words, when a sphere is placed in the space or three-dimension space, then the total space acquired by the sphere generally shows the volume of the sphere. The mathematical formula needed to determine the volume of a sphere is shown below,

[tex]V = \frac{4}{3} \pi r^3[/tex]

Given- The initial radius of the sphere is r1 = 3 inch and the final radius of the sphere is r2 = 3.1 inch.

Now , the change in the radius of the sphere is [tex]dR =[/tex][tex]r2 - r1 =\\[/tex] 3.1 - 3 = 0.1 inch.

The mathematical expression of the volume of a sphere is shown below,

[tex]V = \frac{4}{3}\pi r^3[/tex]   (here, [tex]r[/tex] is the radius of sphere)

Differentiate the above shown expression with respect to radius ([tex]r[/tex]) as shown below,

[tex]\frac{d}{dr} (V) = \frac{d}{dr} (\frac{4}{3} ) \pi r^3[/tex]

∴

dV/dR = 4/3 π × d/dR(r^3)

          = 4/3 π × 3r^2

dV/dR = 4π r^2

∴ dV = 4π r^2 dR            ................(1)

Substitute all values in equation (1) and calculate the approximation as shown below,

dV =  4π r^2 dR

     =  4π (3)^2 × (0.1)

∴ dV = 3.6π cubic inches.

The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.

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The measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.

What is Angle Sum Property?

The sum of all angles of a triangle is equal to the angle of a straight line i.e. 180°. If we have a triangle ABC, then the Sum of angles A , B, and angle C will be 180 ° and the value of the exterior angle is equal to the sum of two interior opposite angle.

We have,

The measures of the angles of a triangle are:

x,

76⁰,

41⁰.

so, we have to calculate the value of the x⁰:

The sum of all the angles in the triangle is 180⁰

so,

to calculate the measure of x put the all values in equation such that using the above rule:

x + 76⁰ + 41⁰ = 180⁰

x = 180⁰ -  76⁰ - 41⁰

x =  63⁰

Hence, the measures of the angles of a triangle are 76⁰, 41⁰, and 63⁰.

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The function y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5 the real number is [tex]\frac{1}{6}[/tex].

y= f(x)= [tex]x^{3}+3x-1[/tex] and a= -5.

The slope of inverse functions are reciprocals at their corresponding points that is,

[tex]f^{-1}'(a)=\frac{1}{f'(b)}[/tex]

Where, [tex]f^{-1}'(a) = b[/tex] and f(b) = a

Now, determine the value of b for a= -5 using f(x)=[tex]x^{3}+3x-1[/tex]

f(b) = a

[tex]b^{3}+3b-1=-5[/tex]

[tex]b^{3}+3b= -4[/tex]

[tex]b = -1[/tex]

Therefore, [tex]f^{-1}'(a) = \frac{1}{f'(b)}[/tex]

[tex]f^{-1}'(-5)=\frac{1}{f'(-1)}[/tex]

Now, find f'(x) and evaluate it at

x= -1

f(x)=[tex]x^{3}+3x-1[/tex]

[tex]f'(x)= 3x^{2}+3[/tex]

f'(x)= [tex]3(x^{2} +1)[/tex]

then, f'(-1)= 3(1+1) = 6

Therefore, [tex]f^{-1}(-5)=\frac{1}{f^{-1}(-1)}=\frac{1}{6}[/tex].

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The expected value E(X) is 7 and the value of variance of X is 5.8333.

Two dice are rolled. Let X be the random variable measuring the sum of the two numbers rolled.

a) The probability of mass function is obtained below:

The possible outcomes in each of the dice are 1 to 6. Therefore, the possible outcomes when two dice is rolled is 36

The sample space, s for fair dice (red die and blue die) is given below:

N(s) = 36

From the given information, two dice are rolled let X be the random variable measuring the sum of the two numbers rolled.

b) The expected value is calculated below:

The probability mass function of X is,

The required mean is,

E(x) = ∑xP(X=x)

[tex]=[2[/tex]×[tex]\frac{1}{36}+3[/tex]×[tex]\frac{2}{36}[/tex]+....+11×[tex]\frac{2}{36}+12[/tex]×[tex](\frac{1}{36} )[/tex]]

=[0.0556+0.1667+0.3333+0.5556+0.8333+1.1111+1.000+0.8333+0.6111+0.3333

=7

C) The variance V(X) is calculated below:

The probability mass function of X is,

The required variance is,

V(x)= ∑[tex](X-x)^{2}P(x) = 5.8333[/tex]

Therefore, the expected value E(X) is 7 and the value of variance of X is 5.8333.

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For the normal distribution of helium porosity (in percentage) of coal samples,

a) 95% CI for the true average porosity of a certain seam is equals to ( 4.56 , 5.14 ).

b) A 98% CI for true average porosity of another seam based on 14 specimens is equals to (4.11 , 5.01).

c) Sample size with width of confidence interval 0.42 is 45.

d) Necessary Sample size to estimate true average porosity to within 0.24 is 60.

We have Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed.

Standard deviations, σ = 0.72

a)Average porosity of specimens , X-bar = 4.85

Sample size, n = 23

Confidence level , alpha = 95% = 0.95

From Normal Distribution table

Z for 95% Confidence Interval = 1.96

so,95% Confidence interval = (x-bar - Z× σ/sqrt(n) , x-bar + Z×σ/sqrt(n) )

plugging all known values in above formula,

= ( 4.85-1.96×0.72/sqrt(23) , 4.85+1.96×0.72/sqrt(23) )

So, 95% Confidence interval = ( 4.56 , 5.14 )

b)Now, Sample size, n = 14

sample mean, X-bar = 4.56

from normal distribution Z- table

Z-score for 98% CI is equal to 2.33

so,98% CI = (x-bar-Z×σ/sqrt(n) , x-bar+Z×σ/sqrt(n) )

= ( 4.56-2.33×0.72/√14 , 4.56+2.33×0.72/√14 )

98% CI = ( 4.11 , 5.01 )

c)Using the normal Distribution Z- table

Z for 95% CI = 1.96

Width of confidence interval = 0.42

we have to determine the sample size, n .

Using formula, Width = 2×Z×σ/sqrt(n)

=> n = (2×Z×sd/width)²

=> n= (2×1.96×0.72/0.42)²

=> n = 45.15

=> n = 45 ( whole number)

d) From normal distribution table

Z for 99% CI = 2.58

Margin of error, ME = 0.24

margin of error = ME = Z×σ/sqrt(n)

=> n = (Z× σ/ME)²

=> n= (2.58×0.72/0.24)²

=> n = 59.9

=> n = 60

Hence, required sample size is 60.

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The distribution of Z = X/Y is  λ₁ / (λ₁+λ₂).

The probability distribution of random variables is described using the cumulative distribution function. The probability for a discrete, continuous, or mixed variable may be described using it. The cumulative probability for a random variable is calculated by adding the probability density function.

Given:

X and Y are independent exponential random variables with respective parameters λ₁ and λ₂.

To find the distribution of Z = X/Y:

First, we have

[tex]f_x_y[/tex](x,y) = λ₁λ₂[tex]e^{-\lambda_1x[/tex][tex]e^{-\lambda_2y[/tex]

First, we find the cumulative distribution function (CDF) for Z = X/Y.

Derivative of Z, f(z) and put  a = x/y

[tex]F_z[/tex](a)  = P (X/Y ≤ a)

        = P (X ≤ aY)

        = [tex]\int\limits^{\infty}_ {x} \ \int\limits^{ay}_0 {\lambda_1\lambda_2e^{-\lambda_1x}e^{-\lambda_2y}\ dx dy[/tex]

       [tex]= \int\limits^{\infty}_ {0} {\lambda_1\lambda_2e^{-\lambda_2y}\ dy [ -1/{\lambda_1}{e^{-\lambda_1x}]\limits^{ay}_ {0}[/tex]

       [tex]= \lambda_2\int\limits^{\infty}_ {0} {e^{-\lambda_2y} - e^{-y(\lambda_2 + \lambda_1a)} \ dy[/tex]

        [tex]= \lambda_2[ [{1/{\lambda_2}+{\lambda_1a]-{e^{-\lambda_2y} + e^{-y(\lambda_2 + \lambda_1a)}]\limits^{\infty}_ {0}[/tex]

        [tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1) - 1][/tex]

        [tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1a) - 1][/tex]

       [tex]= [({\lambda_1a}/{\lambda_2 + \lambda_1a)][/tex]

So, P(X<Y) = P (X/Y <1)

                 = λ₁ / (λ₁+λ₂)

Therefore, distribution is λ₁ / (λ₁+λ₂).

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Answer: -32

Step-by-step explanation:

To solve the equation -128 = 4x for x, we need to isolate the x variable on one side of the equation. We can do this by dividing both sides of the equation by 4. This gives us:

-128/4 = 4x/4

Dividing -128 by 4 gives us -32, so we have:

-32 = 4x/4

We can simplify this by dividing both sides by 4/4 to get:

-32 = x

Thus, the solution to the equation -128 = 4x is x = -32.

The amount of sugar, in ounces, that is needed to make 28 ounces of hummingbird nectar is 7 ounces.

Hummingbird nectar is said to be made by using a ratio of 1 part sugar to 3 parts water. For every 1 ounce of sugar therefore, there would be 3 ounces of water.

Put together, this would give 4 ounces. And the percentage of sugar in this 4 ounces would be:

= 1 / 4

= 25 %

If 28 ounces of hummingbird nectar needed to be made, the sugar needed is :

= Percentage of sugar x Ounces of hummingbird nectar

= 25 % x 28

= 7 ounces of sugar

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Finding the values of d and the value of m with the algebraic working will give us the the value of d to be 7 and the value of m to be 3.

Finding the value of d, we know that the sum of the first n terms of an arithmetic series is given by:

S = n/2 * (2a + (n-1)d)

Since the sum of the first n terms is S and the sum of the first 2m terms is S2m, we can set up the following equation:

S = m/2 * (2a + (m-1)d)

S2m = 2m/2 * (2a + (2m-1)d)

Substituting the given values for S and S2m into these equations, we get:

39 = m/2 * (2a + (m-1)d)

320 = 2m/2 * (2a + (2m-1)d)

Solving for d in each equation, we find that d = -7 in the first equation and d = 7 in the second equation. Since d must be a prime number, the only possible value for d is 7.

Now that we know the value of d, we can solve for m. Substituting the value of d back into one of the equations and solving for m, we get:

39 = m/2 * (2a + (m-1)7)

78 = m * (2a + (m-1)7)

78 = m * 2a + 7m^2 - 7m

7m^2 - m - 78 = 0

We can solve for m using the quadratic formula:

m = (-1 +/- sqrt(1^2 - 4*7*(-78)))/(2*7)

= (-1 +/- sqrt(2521))/14

= (-1 + 49)/14 = 3

= (-1 - 49)/14 = -7

Since m must be a positive integer, the only possible value for m is 3.

Therefore, the value of d is 7 and the value of m is 3.

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The expression that is equivalent to the given expression, (8×10−2) (2.4×10−3), is 1.92 × 10⁻⁴

From the question, we are to determine the expression that is equivalent to the given expression

From the given information,

The given expression is

(8×10−2) (2.4×10−3)

First, we will write this expression properly

The given expression written properly is

(8 × 10⁻²)(2.4 × 10⁻³)

Now, we will evaluate the expression

Evaluating the expression

(8 × 10⁻²)(2.4 × 10⁻³)

8 × 2.4 × 10⁻² × 10⁻³

19.2 × 10⁻⁵

= 1.92 × 10⁻⁴

Hence, the expression is 1.92 × 10⁻⁴

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The answers reasonably approximates the requested value of the sample mean with justification is c. "Bar-w"=13.70 since we can use a normal approximation by the CLT.

Since the sample size is large (n=50), we can use the central limit theorem (CLT) to approximate the sampling distribution of the sample mean with a normal distribution.

The mean of the sampling distribution of the sample mean is equal to the mean of the population, which is 13 feet in this case. The standard deviation of the sampling distribution of the sample mean is equal to the standard deviation of the population divided by the square root of the sample size. This is known as the standard error of the mean, and is denoted by SE(Bar-W).

In this case, SE(Bar-W) = 2.4/sqrt(50) = 0.48.

To find the 98th percentile of a normal distribution with mean 13 and standard deviation 0.48, we can use a standard normal table or a calculator to find that the 98th percentile is approximately 2.05.

Therefore, the value that 98% of samples will have the realized value of Bar-W less than is approximately 13 + 2.05 * 0.48 = 13.70. This means that the correct answer is (c) "Bar-w"=13.70 since we can use a normal approximation by the CLT.

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An airline that wishes to measure customer satisfaction chooses ten of its flights at random over the course of a single month and requests that each passenger fill out a survey. A Cluster sample looks like this.

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Answer: 11.35

Step-by-step explanation: The residual for the rideshare cost with a distance of 16 miles can be calculated using the equation of the least-squares regression line and the observed fare for that distance. The residual is defined as the difference between the observed value and the predicted value.

To find the predicted fare for a distance of 16 miles, we can substitute 16 for x in the equation of the least-squares regression line: ý = 5.20 + 2.33x. This gives us ý = 5.20 + 2.33 * 16 = 42.49.

The observed fare for a distance of 16 miles is 53.84, so the residual is 53.84 - 42.49 = 11.35. Therefore, the answer is 11.35.

Answer:

A. Reflection

Step-by-step explanation:

I believe the answer would be reflection

Answer: 2 times 2 times 2 times 2 times 3 times 3

Step-by-step explanation:

Answer:

the answer is 10 degrees Celsius

50 Degrees Fahrenheit is 10 degrees Celsius.

Here is the formula

(50°F − 32) × 5/9 = 10°C

Answer:

21 m/ph

Step-by-step explanation:

2016 divided by 96

The balloon travels 21 miles in one hour.

In mathematics, division is one kind of operation. The phrases or numbers in this process are split into the same number of parts.

Given, a hot air balloon travels 2016 miles in 96 hours.

And the balloon travels the same number of miles each hour.

To find the same numbers of miles per hour:

We use the division of 2016 by 96.

That means,

2016 / 96

= 21

Therefore, the travel time of the balloon is 21 miles per hour.

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This sequence is not convergent because the nth term does not approach a single limit as n approaches infinity. The limit does not exist, so the answer is DNE.

This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. This means that the value of an does not approach a single limit as n approaches infinity, and so the limit does not exist. Therefore, the answer for the limit as n approaches infinity is DNE.

This sequence does not converge to a single limit because the nth term of the sequence, an, has a coefficient of n, which means that the value of an increases without bound as n increases. As n increases, the value of an increases, and so the value of an does not approach a single limit as n approaches infinity. This means that the limit does not exist, and so the answer for the limit as n approaches infinity is DNE.

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The fifth percentile of the standard normal distribution is -1.64.

In the given question, we have to determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.

The standard normal distribution, commonly known as the z-distribution, is a unique type of normal distribution in which the mean and standard deviation are both equal to 1. Any normal distribution's values can be transformed into z scores to normalise it. Z score shows the amount of standard deviations from the mean that each consists of numbers.

Since, X is normally distributed.

P(X< c) = 0.05

Using Z table

c = -1.645

Fifth percentile = -1.64

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∠8 is the corresponds with ∠3.

Hence, option B is the correct answer.

Congruent angles are those that correspond. Corresponding pairs are all angles that are positioned in relation to the parallel and transversal lines in the same way. When two parallel lines are intersected by another line, comparable angles are the angles that are created in matching corners or corresponding corners with the transversal (i.e. the transversal).

we know that

m∠3=m∠8 --------> by corresponding angles

m∠7=m∠8 -------> by vertical angles

so

m∠3=m∠7 -------> by alternate interior angles

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NOTE: The given question is incomplete on the portal. Here is the complete question.

QUESTION: Which angle corresponds to ∠3?

A) ∠6

B) ∠8

C) ∠1

D) ∠7

a.) The amount that Sammy will have in his account after 10 years will be= $2,165.8

$2,165.8b.) The value of 'r' required would be = 0.27%

Interest rate is defined as the rate at which an individual receives an amount of money from an investment made over a period of time.

The amount invested by Sam(P) = $1700

The interest rate(R) = 2.74%

The time of investment (T) = 10 years

Simple interest = P×T×R/100

= 1700×10×2.74/100

= 46580/100

= $465.80

Therefore, the amount that Sammy will have in his account after 10 years will be = $1700 + $465.80

= $2,165.8.

For David's account interest rate the following is carried out:

The amount invested by Sam(P) = $1700

The interest rate(R) = r%

The time of investment (T) = 10 years

Simple interest (SI) = $465.80

Using the formula for simple interest;

SI = P×T×R/100

Make R the subject of formula;

R = SI×100/P×T

R = 465.80×100/1700×100

R= 46580/170000

R= 0.27%

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Complete question:

Sam invests $1700 in a saving account that pays a nominal annual rate of interest of 2.74% compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account. (a) Find the amount that Sam will have in his account after 10 years.

David also invests $1700 in a savings account that plays an annual rate of interest of r%, compounded yearly. David makes no further payments or withdrawals from this account. (b) Find the value of 'r' required so that the amount in David's account after 10 years will be equal to the amount in Sam's account.

Answer:

Area = 12a m²

Step-by-step explanation:

Given information,

→ Length = 4a m

→ Width = 3a m

Now we have to,

→ find the area of rectangular park.

Formula we use,

→ Area = L × W

Then the area of rectangle is,

→ L × W

→ 4a × 3a

→ (4 × 3)a

→ 12a m²

Therefore, the area is 12a m².

Option C, The research situation that would be most likely to use an independent-measures design is to compare the mathematics skills of 9th-grade boys versus 9th-grade girls.

An independent-measures design is used when there are two or more groups that are being compared and each individual only belongs to one group. In this research situation, the two groups being compared are 9th-grade boys and 9th-grade girls. Each individual in the study belongs to one group or the other, and they are not part of both groups. This makes it an ideal situation for using an independent-measures design.

Option a) examine the development of vocabulary as a group of children mature from age 2 to age 3 would likely use a within-subjects design, as the same group of children is being measured at two different time points.

Option b) Examining the long-term effectiveness of a stop-smoking treatment by interviewing subjects 2 months and 7 months after the treatment ends would also likely use a within-subjects design, as the same group of subjects is being measured at two different time points.

Option d) compare the blood-pressure readings before medication and after medication for a group of patients with high blood pressure would likely use a repeated-measures design, as the same group of patients is being measured at two different time points and the order of the measurements (before or after medication) is being controlled.

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Answer:

-5/8

Step-by-step explanation:

5.(-2^-3)

-5. 1/2³

-5. 1/8

-5/8

On solving the provided question, From the question, our region is defined by: lower bound: y= [tex]x^2[/tex] and upper bound: y = [tex]\sqrt{x}[/tex]

Integrals are mathematical representations of numbers and functions that express notions such as volume, area, displacement, and other outcomes of the combination of little data. Finding integrals is the term used to describe the procedure.

By green's theorem -

[tex]\int\limits^{}_{} {} \, \int\limits^{}_{a} {5dA} \, = 5/3[/tex]

First, the integral given in this exercise corresponds to:  

[tex]\int\limits^{}_{C} {((3y + 7e\sqrt{x}dx) +( 8x+ 5cos(y^2)) } \, dy[/tex]

Greens Theorem given as,

[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {(-\beta /\beta _{y} )P(x,y) + (\beta /\beta _{y} )Q(x,y)} \, dA[/tex]

and we have -

P(x,y) = 3y + [tex]7e^{\sqrt{x}}[/tex]

Q(x,y) = 8x + 5cos([tex]y^2[/tex])

And,

[tex]\int\limits^{}_{C} {(P(x,y)dx +Q(x,y)dy)} \, = \int\limits^{}_{} {x} \, \int\limits^{}_a {5dA} \, dA[/tex]

From the question, our region is defined by:

lower bound: y= [tex]x^2[/tex]

upper bound: y = [tex]\sqrt{x}[/tex]

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Let x be the length of the shorter wall of the rectangular room and y be the length of the longer wall of the rectangular room. Since the longer wall is 7 ft longer than the shorter wall, we can write the equation y = x + 7. We can also express the diagonal of the rectangular room using the Pythagorean theorem as x^2 + y^2 = 13^2. We can solve for x and y by substituting the equation y = x + 7 into the equation x^2 + y^2 = 13^2 and then solving for x. Doing this, we get x^2 + (x + 7)^2 = 13^2. Expanding the square on the right side of the equation and then rearranging the terms, we get x^2 + 2x^2 + 14x + 49 = 169. Combining like terms, we get 3x^2 + 14x - 120 = 0. This quadratic equation can be factored as (x - 8)(3x + 15) = 0. Since the length of a side of a rectangle must be positive, we can ignore the solution x = -15/3. So, the length of the shorter wall of the rectangular room is x = 8 ft. The length of the longer wall can be found by substituting this value into the equation y = x + 7, giving us y = 8 + 7 = 15 ft. Therefore, the dimensions of the rectangular room are 8 ft by 15 ft.

The solution is Option B , Option D , Option E

The athletes with speeds 20m/s and 30m/s will meet at every minute

What is Speed?

Speed is defined as the rate of change of position of an object in any direction. Speed is measured as the ratio of distance to the time in which the distance was covered. Speed is a scalar quantity as it has only direction and no magnitude

Speed = Distance / Time

Given data ,

Let the equation be represented as A

Now , the value of A is

The distance of the track is D = 600 m

Let the first athlete be A

Let the second athlete be B

The speed of A = 20 m/s

The speed of B = 30 m/s

The time taken by A to complete 600 m = 600 / 20 = 30 seconds

The time taken by B to complete 600 m = 600 / 30 = 20 seconds

Now , the time at which A and B will cross each other is the LCM of their respective time

So , The LCM of 20 and 30 is = 60 seconds

Therefore , the athletes will meet together at every 60 seconds or every 1 minute

Hence , the athletes will meet at every minute

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James owns just 5% of the company after the investment.

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100.

Percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%".

Percentage can be calculated by dividing the value by the total value, and then multiplying the result by 100. It is given by:

Percentage = (value / total value) * 100%

How to find the Percentage owned by James;

James' investment: $10,000

The other 3 people invested;

x = $10,000

y = $10,000

z = $10,000

Total investment made was = $200,000

James' Percentage = [tex]\frac{10000}{200000\\}[/tex] x 100

James' Percentage = 5%

So, we can say that James owns 5% of the total investment in the business.

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The graph of x-3y = 12 is given.

A graph is a visual representation or diagram that displays facts or values in an organized manner in mathematics. The points on a graph are typically used to depict the relationships between two or more things.

The equation is

x-3y = 12.

The graph of this line intersects the x-axis and y-axis.

And end behavior is infinity on both positive and negative.

And the graph is a straight line graph.

Therefore, behavior of x -3y = 12 is given in the graph.

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Answer:

  x = 16 1/3°

Step-by-step explanation:

You have parallel lines with consecutive interior angles marked 7x-28° and 5x+12°, and you want to find the value of x.

At parallel lines, consecutive interior angles, or same-side interior angles, are supplementary. This means the total of the two marked angles is 180°.

  7x -28° +5x +12° = 180°

  12x -16° = 180° . . . . . . . . . . simplify

  12x = 196° . . . . . . . . . add 16°

  x = (49/3)° = 16 1/3° . . . . . . . divide by 12

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Additional comment

∠3 = 86 1/3°

∠5 = 93 2/3°