A 25-foot ladder is placed against a vertical wall of a building, with the bottom of the ladder standing on level ground 20 feet from the base of the building. How high up the wall does
the ladder reach? Show work.

The value of x in the equation 2x³ + 3x² + 12x + 1 = 0​ when solved is x = -0.085

Given the following equation

2x³ + 3x² + 12x + 1 = 0​

The equation cannot be factorized

So, we solved graphically

See attachment for the graph of the equation 2x³ + 3x² + 12x + 1 = 0​

On the graph, we observe that the curve of the equation crosses the x-axis at

x = -0.085

This means that the solution for x in the equation is x = -0.085

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The equation that translates the information is $282 = $150 + $6x where x ​ to represent the number of guests at your party

The total charge for renting a roller rink for a birthday party is composed of two parts: a fixed cost of $150 and a variable cost of $6 per person. The variable cost depends on the number of guests, which we can represent with the variable x.

Therefore, the total cost equation can be written as:

Total cost = Fixed cost + Variable cost

Total cost = $150 + $6x

Here, Total cost = $282

So, equation is $282 = $150 + $6x

To find the total charge for a specific number of guests, we can substitute the value of x into the equation and simplify it. For example, if there were 22 guests at the party, the total cost would be:

Total cost = $150 + $6(22)

Total cost = $150 + $132

Total cost = $282

This means that the total charge for a party with 22 guests would be $282.

It's important to note that the fixed cost of $150 is the cost that remains the same no matter how many guests attend the party. On the other hand, the variable cost of $6 per person changes based on the number of guests.

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Janelle calculated that the unknown weighed lost 3.219 grams of mass when exposed to a flame.

Janelle weighed an unknown substance to determine its mass. The initial mass was 20. 609 grams. She then exposed the substance to a flame, and carefully weighed it again. This time, the mass was 17. 39 grams. To determine the amount of mass x the substance lost, Janelle can write and solve an equation. The equation would be:  20.609 - 17.39 = x, where x is the amount of mass lost. To solve for x, Janelle can subtract 17.39 from 20.609. This calculation would result in x = 3.219, meaning the substance lost 3.219 grams of mass. Therefore, the amount of mass x the substance lost is 3.219 grams.

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The Coefficient of determination i.e. r²≈ 0.9155.

What is Coefficient of determination?

The coefficient of determination, denoted as r², is a statistical measure that represents the proportion of the variance in the dependent variable that is predictable from the independent variable(s) in a regression model.

To determine the coefficient of determination, we first need to calculate the regression sum of squares (SSR) and the total sum of squares (SST) using the following formulae:

SSR = Σ(y_pred - y_mean)²

SST = Σ(y - y_mean)²

where y_pred is the predicted value of y based on the regression equation, y_mean is the mean of y, and y is the actual value of y.

Once we have calculated SSR and SST, we can use the formula for the coefficient of determination:

r² = SSR / SST

Using the given values, we have:

SSR = 131,742.28

SST = 143,920.65

r² = SSR / SST = 131,742.28 / 143,920.65 ≈ 0.9155

Interpreting the meaning of r², we can say that approximately 91.55% of the variation in the cost of a meal can be explained by the variation in summated ratings. In other words, the model based on summated ratings is able to explain a large proportion of the variability in the cost of a meal.

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Answer:

5/6

Step-by-step explanation:

The length of the value x in the right triangle is 20.7 units.

A right triangle is a triangle that has one of its angles as 90 degrees. The sum angles in a triangle is 180 degrees.

Therefore, the side of a right triangle can be found using trigonometric ratios as follows:

Hence,

cos 34° = adjacent / hypotenuse

Therefore,

adjacent side = x

hypotenuse side = 25

Hence,

cos 34 = x  /25

cross multiply

x = 25 cos 34

x = 25 × 0.82903757255

x = 20.7259393139

Therefore,

x = 20.7 units

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Answer: It is rational

Step-by-step explanation:

Answer:

To compare the slopes of Function A and Function B, we need to find the slope of each function.

Function A: y = 104x

The slope of a linear function in the form y = mx + b is equal to the coefficient of x, which is 104 in this case. Therefore, the slope of Function A is 104.

Function B: y = -x + 7

The slope of a linear function in the form y = mx + b is equal to the coefficient of x, which is -1 in this case. Therefore, the slope of Function B is -1.

Since 104 is greater than -1, the statement "The slope of Function A is greater than the slope of Function B" is true.

Therefore, the answer is: The slope of Function A is greater than the slope of Function B.

Answer:

x = 3√7.

y = 12.

Step-by-step explanation:

The 2 triangles are similar so:

x/9 = 7/x

x^2 = 63

x = √63

   = 3√7.

By Pythagoras' theorem

y^2 = 9^2 + (3√7)^2

      =  81 + 63

      = 144

y = 12.

Answer:

Step-by-step explanation:44,4444444

Answer:

0.0225

Step-by-step explanation:

18÷800=0.0225

Answer:

Below

Step-by-step explanation:

With questions like this it is helpful to draw the scenario as in the image below.     Using Pyhtagorean theorem , you can calculate 'x'..... then the highway distance in range is two times this = 12.6 km  

Q1 . By pytagorean therom ,

CA ^ 2 + AB ^2 = BC^2

2^2 +4^2 = BC^2

4 + 16 = BC ^2

20 = BC^2

BC = [tex]\sqrt{20}[/tex]

BC = 4.4

1/2 BC = x = 2.2 ( Midpoint)

Q2 .

OQP = 90

(5+X)^2 = X^2 + 7^2

25 + 10X + X^2 = X^2 + 49

25 + 10X = 49

10X = 49 - 25 = 24

X = 2.4

Answer:

see explanation

Step-by-step explanation:

(a)

the angle in a semicircle is a right angle , then Δ ABC is right with ∠ BAC = 90°

the radius of the circle OB = x , then BC = 2x

using Pythagoras' identity in the right angle

BC² = AB² + AC²

(2x)² = 4² + 2²

4x² = 16 + 4 = 20 ( divide both sides by 4 )

x² = 5 ( take square root of both sides )

x = [tex]\sqrt{5}[/tex]

-----------------------------------------------------------------

the angle between a tangent and the radius of the circle at the point of contact is 90° , that is

∠ OQP = 90° and Δ OPQ is right

note that OP = 5 + radius = 5 + x

using Pythagoras' identity in the right triangle

OP² = OQ² + PQ²

(5 + x)² = x² + 7² ← expand left side using FOIL

25 + 10x + x² = x² + 49  ( subtract x² from both sides )

25 + 10x = 49 ( subtract 25 from both sides )

10x = 24 ( divide both sides by 10 )

x = 2.4

Hence, in response to the provided question, we can say that the constant k denotes the graph's vertical shift. The coefficient "a" specifies the function's opening direction and amount.

Mathematicians research numbers and their variants, equation and related structures, objects and their locations, and prospective locations for these things. The term "module" is used to describe the connection that exists in between set of inputs, each of which has a corresponding output. A function is an input-output connection in which each inputs results in something like a single, distinct return. A domain, codomain, or scope is assigned to each function. Functions are usually denoted by the letter f. (x). An x is used for entry. On capabilities, one-to-one capabilities, multiple prowess, in capabilities, and on capabilities are the four major types of accessible functions.

False.

A quadratic function has the usual form [tex]f(x) = a(x - h)^2 + k[/tex], where "a," "h," and "k" are constants that determine the shape, position, and orientation of the quadratic function's graph. The phrase (x - h)2 denotes the squared distance between the input value x and the horizontal shift h, while the constant k denotes the graph's vertical shift. The coefficient "a" specifies the function's opening direction and amount.

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To rewrite a problem with negative exponents into a problem with positive exponents, you can use the following rules.

Move any negative exponent to the denominator (or numerator, depending on where it is located) by changing the sign of the exponent to positive.

If there are any fractions with negative exponents, flip the fraction so that the exponent becomes positive.

For example, if you have the expression:

[tex]X^-2 / y^-3[/tex]

To rewrite this with positive exponents, you can move the negative exponents to the denominator and change the signs:

[tex]y^3 / x^2[/tex]

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Assuming that David sold the stock at $10 per stock today, then, we will agree that he bought the Oracle stock yesterday at a price of approximately $9.89 per share.

A share, also known as a stock or equity, is a unit of ownership in a corporation. Stock investments provide investors with returns in the form of dividends and capital gains. The higher the expected returns, the more shares an investor owns.

Let the price at which David bought each share of Oracle stock yesterday be x. Then, his total cost for buying 200 shares would be: 200x

Similarly, his total revenue for selling 200 shares at $10 per share today would be:

= 200*($10)

= $2000

His profit was $22, so we can set up the equation which is $2000 - 200x = $22.

Simplifying and solving for x, we get:

$1978 = 200x

x = $1971/200

x = $9.89.

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ln(y) = -0.544x + 1.099

The logarithmic equation that relates x and y is given by ln(y) = -0.544x + 1.099. To get this equation, the given table of values is considered.Let's first find the value of "a" in the general logarithmic equation, which is y = a ln(x). We use the given table of values to get this value. When x = 1, y = 1, and so a ln(1) = 1, which implies that a = 1. Hence, the equation becomes y = ln(x).Now, we need to convert this to an equation in terms of y and x. That is, we need to find ln(y) in terms of x. To do this, we substitute y for x in the equation y = ln(x). That is, we get ln(y) = ln(x).Now, we substitute the given values of x and y in this equation to get the required logarithmic equation. That is, we have:ln(y) = -0.544x + 1.099 (rounded to three decimal places). Therefore, the logarithmic equation that relates x and y is ln(y) = -0.544x + 1.099.

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Mean = 110

Standard Deviation = 17

The solution is given below:A. What is the probability that X is greater than 138.05?We have, Mean (μ) = 110Standard Deviation (σ) = 17Let's first standardize X = 138.05: Z = (X - μ)/σZ = (138.05 - 110)/17 = 1.650Therefore, P(X > 138.05) = P(Z > 1.650)Using a standard normal table, the probability of Z being greater than 1.650 is:0.0495Probability = 0.0495B. What value of X does only the top 15% exceed?We have, Mean (μ) = 110Standard Deviation (σ) = 17We need to find the value of X such that only the top 15% exceed. In other words, we want to find X such that P(X > x) = 0.15.Using a standard normal table, we can find that the Z-value for the top 15% is 1.0364. We can write this as:P(Z > 1.0364) = 0.15We now standardize the equation:P(Z > 1.0364) = 0.15Z = invNorm(0.15) + 1.0364Z = -1.0364 + 1.0364Z = 0Therefore, we have:Z = (X - μ)/σ0 = (X - 110)/17X = 110Therefore, the value of X such that only the top 15% exceed is 110.

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A. The probability that X is greater than 138.05 is 0.0492.

B. The value of X does only the top 15% exceed is 127.57.

Given that X is normally distributed with mean 110 and standard deviation 17.

A. Probability= P(X > 138.05)

First, we need to calculate the z-score of 138.05 by using the formula;

z = (X - μ)/σ

Where

X = 138.05, μ = 110, and σ = 17z = (138.05 - 110)/17 = 1.649

Therefore, P(X > 138.05) = P(Z > 1.649)

Using a standard normal table, the probability P(Z > 1.649) = 0.0492

Thus, the probability that X is greater than 138.05 is 0.0492.

B. We know that the mean of the normal distribution is 110 and the standard deviation is 17, and we need to find the value of X such that only the top 15% exceed.

Therefore, we need to find the value of X such that P(X > x) = 0.15

Using a standard normal table, we find that the Z-value of 0.15 is 1.0364

Therefore, z = (x - μ)/σ = 1.0364

Solving for x, we get,

x = σz + μ = 17 × 1.0364 + 110 = 127.57

Thus, the value of X that only the top 15% exceed is 127.57.

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The mean of Y is -5/4 and the variance of Y is 144.

In statistics, the mean is a measure οf central tendency that represents the average οf a set οf data. It is alsο knοwn as the arithmetic mean. The mean is calculated by adding up all the values in the dataset and dividing the sum by the tοtal number οf values. It is οften used tο describe the typical οr average value in a set οf data.

Variance is a statistical measure that describes hοw much the values in a set οf data vary frοm the average value οr mean. It measures the spread οr dispersiοn οf the data pοints arοund the mean.

In the given questiοn,

Tο find the mean and variance οf the randοm variable Y, we need tο use the fοrmulas fοr the expected value and variance οf a functiοn οf a randοm variable:

The expected value (mean) of Y is given by:

E(Y) = E(3X - 2) = 3E(X) - 2

The variance of Y is given by:

Var(Y) = Var(3X - 2) = 9Var(X)

To find E(X), we need to integrate the density function f(x) over all possible values of X:

E(X) = ∫0∞ x f(x) dx

= ∫0∞ x (1/4)[tex]\rm e^{(-x/4)[/tex]dx

This integral can be solved using integration by parts, with u = x and dv/dx = (1/4)[tex]\rm e^{(-x/4)[/tex] dx.

Integrating by parts, we get:

E(X) = [-x/4 [tex]\rm e^{(-x/4)[/tex]]_0∞ + ∫0∞ (1/4)[tex]\rm e^{(-x/4)[/tex] dx

= [0 + (1/4)]/1 + [0]

= 1/4.

Therefore, the expected value of Y is:

E(Y) = 3E(X) - 2 = 3(1/4) - 2 = -5/4.

To find Var(X), we can use the formula for the variance of an exponential distribution, which is:

Var(X) = (1/λ²) = (4²) = 16.

Therefore, the variance of Y is:

Var(Y) = 9Var(X) = 9(16)

= 144.

Therefore, the mean of Y is -5/4 and the variance of Y is 144.

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The mean of Y is 10 minutes, and the variance of Y is 144 minutes squared.

What is integration?

Integration is a mathematical operation that is the reverse of differentiation. Integration involves finding an antiderivative or indefinite integral of a function.

To find the mean and variance of Y, we will first need to find the mean and variance of X.

Mean of X:

The mean of X is given by:

E(X) = integral from 0 to infinity of xf(x) dx

E(X) = integral from 0 to infinity of x * 1/4 e^(-x/4) dx

Using integration by parts with u = x and dv = e^(-x/4) dx, we can evaluate this integral to get:

E(X) = 4

Variance of X:

The variance of X is given by:

Var(X) = E(X^2) - [E(X)]^2

We will first need to find E(X^2). Using integration by parts again, we can evaluate this integral as:

E(X^2) = integral from 0 to infinity of x^2 * 1/4 e^(-x/4) dx

E(X^2) = 32

Therefore, the variance of X is:

Var(X) = E(X^2) - [E(X)]^2

Var(X) = 32 - 16

Var(X) = 16

Now, we can use the formula for the mean and variance of a linear transformation of a random variable to find the mean and variance of Y.

Mean of Y:

The mean of Y is given by:

E(Y) = E(3X - 2)

E(Y) = 3E(X) - 2

E(Y) = 10

Variance of Y:

The variance of Y is given by:

Var(Y) = Var(3X - 2)

Var(Y) = 9Var(X)

Var(Y) = 144

Therefore, the mean of Y is 10 minutes, and the variance of Y is 144 minutes squared.

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Carina needs to hit the ball at a height of about 3.04 feet to clear the net and land in the service box.

What is the trigonometric ratio?

There are six trigonometric ratios- sine(sin), cosine(cos), tangent(tan), cotangent(cot), secant(sec), cosecant(cosec). Let θ be the angle of the right-angled triangle then, sin θ = opposite side/hypotenuse or 1/cosθ.

To determine the height Carina needs to hit the ball to clear the net and land in the service box, we can use the following steps:

Draw a bird's-eye view diagram of the tennis court, including the net, the service box, and the path of the ball.

Draw a side view diagram of Carina and the ideal path of the tennis ball, including the measurements that are needed from the bird's-eye view diagram.

Use trigonometry to calculate the height Carina needs to hit the ball.

Determine whether it is reasonable for Carina to reach this height, given her height of 5'7".

Calculate the angle at which the ball hits the ground.

Here are the steps in more detail:

Bird's-eye view diagram:

We can draw a bird's-eye view diagram of the tennis court, with the net located at the center and the service box 21 feet away from the net. The distance from the net to the service box is (78-21)/2 = 28.5 feet. We can label the diagram with these distances and the height of the net (3 feet).

Side view diagram:

Next, we can draw a side view diagram that shows Carina, the ideal path of the tennis ball, and the measurements that are needed from the bird's-eye view diagram. We can label the diagram with the height of the tennis racket (h) and the distance from the racket to the net (x). We also need to label the height Carina needs to hit the ball to clear the net (y) and the height of the net (3 feet).

Calculate the height Carina needs to hit the ball:

To calculate the height Carina needs to hit the ball, we can use the following trigonometric formula:

tan(θ) = y / x

where theta is the angle between the ideal path of the ball and the ground, y is the height Carina needs to hit the ball, and x is the distance from the racket to the net.

Solving for y, we get:

y = x * tan(θ)

We know that x = 21 feet, and we can estimate that the angle theta is about 8 degrees (since the ball needs to clear the net by at least 3 feet). Plugging in these values, we get:

y = 21 * tan(8) = 3.04 feet

So Carina needs to hit the ball at a height of about 3.04 feet to clear the net and land in the service box.

Determine whether it is reasonable for Carina to reach this height:

Carina's height is 5'7", which is equivalent to 67 inches.

If we assume that Carina's arm reaches to about shoulder height, which is roughly halfway between her height and the height she needs to hit the ball (3.04 feet), then she would need to hit the ball about 1.5 feet above her shoulder.

This seems like a reasonable height for a skilled tennis player.

Calculate the angle at which the ball hits the ground:

To calculate the angle at which the ball hits the ground, we can use the following trigonometric formula:

tan(θ) = y / (x + d)

where d is the distance from the net to where the ball lands. We know that y = 3.04 feet, x = 21 feet, and d = 21 + 21 = 42 feet (since the ball travels the same distance on the other side of the court). Plugging in these values, we get:

tan(θ) = 3.04 / 42

θ = arctan(3.04 / 42)

θ = 0.07225

Hence, Carina needs to hit the ball at a height of about 3.04 feet to clear the net and land in the service box.

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Answer:

the answer is 5 simplify

Answer:

20/14

10/7

0r 5/56

Step-by-step explanation:

i hope you find it helpful

59.1%

Step-by-step explanation:

100% / 22 x 13 =59.1%

Answer:

0.62

Step-by-step explanation:

given :
[tex]\frac{2}{5} * 1\frac{4}{7}[/tex]
[tex]=\frac{2}{5} *\frac{11}{7}[/tex]
[tex]= \frac{22}{35}[/tex]
[tex]=0.62[/tex] as a decimal

Hope it helps

To find the amount of the discount, we can multiply the total bill by the discount rate of 20% (or 0.2):

Discount = 0.2 x $54.95

Discount = $10.99

What is percentage ?

Percentage is a way of expressing a fraction or proportion as a portion of 100. It is often used to represent rates, discounts, or increases/decreases in values.

So you would save $10.99 using your Hawk card.

To find the final amount you would pay after the discount, we can subtract the discount from the total bill:

Final amount = Total bill - Discount

Final amount = $54.95 - $10.99

Final amount = $43.96

So the final amount you would pay with the Hawk card discount is $43.96.
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The maximum number of batches of smoothie that can be made from 42 litres of lime juice is 50.

To determine the amount of mixed fruit juice required to make a batch of smoothie, we need to find the total volume of ice cream and mixed fruit juice in a batch.

The ratio of ice cream to mixed fruit juice is 5 : 3, which means that the total ratio of ingredients in a batch of smoothie is 5 + 3 = 8.

Since the batch size is 6.4 litres, the volume of mixed fruit juice required per batch is:

3/8 x 6.4 litres = 2.4 litres

Next, we need to determine the amount of lime juice in each batch. If 35% of the mixed fruit juice is lime juice, then the amount of lime juice in each batch is:

35/100 x 2.4 litres = 0.84 litres

Now we can determine the number of batches of smoothie that can be made from 42 litres of lime juice.

The amount of lime juice required per batch is 0.84 litres, so the maximum number of batches that can be made from 42 litres of lime juice is:

42 litres ÷ 0.84 litres per batch = 50 batches

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Therefore , the solution of the given problem of unitary comes out to be C is the correct response. $94.50.

The task can be finished by combining the data expression obtained to use this nanosection methodology with in all supplemental data from two people who employed a particular variable tactic. In plain English, this indicates that, if the desired result occurs, either the stated individual will be known or, in fact, the colour by both enormous processes would be skipped. A refundable fee of Rupees ($1.01) may be necessary for forty pens.

Here,

By dividing both sides by D, we can determine B if P = DB:

=> B = P/D

Sandra's premium P will be equal to the $567 base premium because her risk factor D is 1.0.

So,

=> B = P/D = $567/1.0 = $567

Her monthly payment will be as follows because this is her six-month premium:

=> Monthly premium = six-month premium / six,

which equals $567 / six, or $94.50.

Thus, C is the correct response. $94.50

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Mrs. Rose receives the greater pay rise of £660 per year compared to Mrs. Patel's pay rise of £645.

To compare the pay rise for Mrs Patel and Mrs Rose, we need to calculate the actual amount of the pay rise for each of them.

For Mrs Patel, her pay rise is 2.5% of her current salary of £25,800:

Pay rise for Mrs Patel = 2.5% of £25,800 = £645

For Mrs. Rose, her pay rise is £55 per month, which is:

Pay rise for Mrs. Rose = £55 x 12 = £660 per year

Comparing the two pay rises, we see that Mrs. Rose receives a greater pay rise of £660 per year, compared to Mrs. Patel's pay rise of £645. Therefore, Mrs. Rose receives the greater pay rise.

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Answer: 35

Step-by-step explanation:

 Let's use Euler's formula to solve this problem. According to Euler's formula, for any convex polyhedron (geometric solid with flat faces), the number of faces (F), vertices (V), and edges (E) satisfy the equation F + V - E = 2.

We are given that the geometric solid has 12 faces and 25 vertices. Let's call the number of edges "E" and substitute these values into Euler's formula:

12 + 25 - E = 2

Simplifying this equation, we get:

E = 35

Therefore, the geometric solid has 35 edges.

Answer:

[tex]\large\boxed{\textsf{x=3}}[/tex]

Step-by-step explanation:

[tex]\textsf{We are asked to find the value of x.}[/tex]

[tex]\textsf{Notice that we have Intersecting Chords inside of the circle.}[/tex]

[tex]\large\underline{\textsf{What are Chords?}}[/tex]

[tex]\textsf{Chords are line segments that have endpoints on the circumference of a circle.}[/tex]

[tex]\textsf{Note that some Chords can be Diameters, but not for this problem. There's no Center.}[/tex]

[tex]\textsf{These Chords don't create Perpendicular Lines, which means the angles aren't equal.}[/tex]

[tex]\textsf{We can identify the unknown angles with a postulate.}[/tex]

[tex]\large\underline{\textsf{What is a Postulate?}}[/tex]

[tex]\textsf{A Postulate is a statement that can never be false. No matter what, it's always}[/tex]

[tex]\textsf{true.}[/tex]

[tex]\textsf{We can use the Linear Pair Postulate to find the needed angle.}[/tex]

[tex]\large\underline{\textsf{What is the Linear Pair Postulate?}}[/tex]

[tex]\textsf{Linear Pair Postulate is a Postulate that proves 2 angles add up to 180}^{\circ}.[/tex]

[tex]\textsf{A Linear Pair is 2 Adjacent Angles that form a Straight Angle.}[/tex]

[tex]\large\underline{\textsf{For our Problem;}}[/tex]

[tex]\tt \angle HLJ \ and \ \angle ILJ \ are \ Linear \ Pairs.[/tex]

[tex]\large\underline{\textsf{Find} \ \angle \textsf{HLJ;}}[/tex]

[tex]\tt \angle HLJ + \angle ILJ = 180^{\circ}.[/tex]

[tex]\tt \angle HLJ + 85.5^{\circ} = 180^{\circ}.[/tex]

[tex]\underline{\textsf{Subtract 85.5 from both sides of the equation;}}[/tex]

[tex]\tt \angle HLJ = 94.5^{\circ}[/tex]

[tex]\textsf{Now that we have} \tt \ \angle HLJ, \textsf{we can find x.}[/tex]

[tex]\textsf{We are given 2 arc measures and we have 2 Intersecting Chords.}[/tex]

[tex]\textsf{We should use} \tt \ \angle HLJ \ \textsf{to find x.}[/tex]

[tex]\large\underline{\textsf{How?}}[/tex]

[tex]\textsf{The Angle is equal to half the sum of the 2 arcs.}[/tex]

[tex]\underline{\textsf{For our Problem;}}[/tex]

[tex]\tt \angle HJL = \frac{1}{2} (42x-6 + 15x+24)[/tex]

[tex]\tt 94.5^{\circ} = \frac{1}{2} (42x-6 + 15x+24)[/tex]

[tex]\large\underline{\textsf{Solve;}}[/tex]

[tex]\textsf{Multiply each side by 2 first.}[/tex]

[tex]\tt 189^{\circ} = 42x-6 + 15x+24[/tex]

[tex]\underline{\textsf{Combine Like Terms;}}[/tex]

[tex]\tt 189^{\circ} = 57x+18[/tex]

[tex]\underline{\textsf{Subtract 18 from both sides;}}[/tex]

[tex]\tt 171^{\circ} = 57x[/tex]

[tex]\underline{\textsf{Divide each side by 57;}}[/tex]

[tex]\large\boxed{\textsf{x=3}}[/tex]

It took Simon 45 minutes to finish the run.

What is distance?

Distance is the measure of how far apart two objects or locations are from each other. It is usually measured in units such as meters, kilometers, miles, or feet.

To solve this problem, we need to use the distance formula:

Distance = Rate x Time

We can break down the run into three parts: the 2 miles due west, the 1.5 miles due north, and the straight line back to the parking lot.

For the first part, Simon ran 2 miles at a rate of 8 miles per hour. Therefore, the time it took him to run this part of the trail was:

Time = Distance / Rate

Time = 2 miles / 8 miles per hour

Time = 0.25 hours

For the second part, Simon ran 1.5 miles at a rate of 8 miles per hour. Therefore, the time it took him to run this part of the trail was:

Time = Distance / Rate

Time = 1.5 miles / 8 miles per hour

Time = 0.1875 hours

To find the distance of the last part, we need to use the Pythagorean theorem to find the hypotenuse of the right triangle formed by the 2-mile and 1.5-mile legs:

c² = a² + b²

c² = 2² + 1.5²

c² = 4 + 2.25

c² = 6.25

c = sqrt(6.25)

c = 2.5 miles

So, Simon ran 2.5 miles at a rate of 8 miles per hour. Therefore, the time it took him to run this part of the trail was:

Time = Distance / Rate

Time = 2.5 miles / 8 miles per hour

Time = 0.3125 hours

Finally, we can add up the times for each part of the trail to find the total time it took Simon to finish the run:

Total Time = 0.25 hours + 0.1875 hours + 0.3125 hours

Total Time = 0.75 hours

To convert this to minutes, we can multiply by 60:

Total Time = 0.75 hours x 60 minutes/hour

Total Time = 45 minutes

Therefore, it took Simon 45 minutes to finish the run.

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The equation that models the relationship between the models to show Ethan's employee benefits is y = 14 + 3(x - 1).

The number of paid vacation days Ethan earned was 35 paid vacation days .

Ethan will reach the maximum number of paid vacation days in 7 years.

Let x represent the number of years he has worked for this employer and y represent the number of paid vacation days he has earned.

The equation that models the relationship between x (years worked) and y (paid vacation days earned) is:

y = 14 + 3(x - 1)

Using the equation from part a, we can find out how many paid vacation days Ethan has earned after eight years:

y = 14 + 3(8 - 1)

y = 14 + 3(7)

y = 14 + 21

y = 35

To determine when Ethan will reach the maximum number of paid vacation days, we can set y equal to 30 and solve for x:

30 = 14 + 3(x - 1)

16 = 3(x - 1)

16/3 = x - 1

x = 16/3 + 1

x = 16/3 + 3/3

x = 19/3

x = 7

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