Help please need this asap

Answer:

Point T is on a line segment SU. Given SU=4x+1, TU=3x, and ST=3x-1, determine the numerical length of SU

Step-by-step explanation:

By the segment addition rule, length of ST + length TU equals length of SU so 3x -1 + 3x = 4x + 1 6x - 1 = 4x + 1 2x = 2 x = 1 substituting value of x back into formula for length of SU: 4 (1) + 1 = 5

The perimeter of the trapezoid is 16 units

The formula used for calculating the perimeter of a trapezoid is expressed with the equation;

P = a + b + c + d

Such that the parameters of the equation are;

We then have;

a = 1 units

b= 9 units

c = 3 units

d = 3

Substitute the values

Perimeter = 1 + 9 + 3 + 3

Add the values

Perimeter = 16 units

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Answer:

First problem: x = √15

Second problem: x = 6√5 (B)

Third problem: x = 6, y = 12 (A)

Fourth problem: x = 6√2 (C)

Step-by-step explanation:

We use geometric means as follows:

First problem:

[tex] \frac{3}{x} = \frac{x}{5} [/tex]

[tex] {x}^{2} = 15[/tex]

[tex]x = \sqrt{15} [/tex]

Second problem:

[tex] \frac{20}{x} = \frac{x}{9} [/tex]

[tex] {x}^{2} = 180[/tex]

[tex]x = \sqrt{180} = \sqrt{36} \sqrt{5} = 6 \sqrt{5} [/tex]

So B is the correct answer.

Third problem:

We have a 30°-60°-90° triangle.

x = 6, y = 12. So A is the correct answer.

Fourth problem:

We have a right isosceles triangle.

x = 12/√2 = 6√2. So C is the correct answer.

The interquartile range (IQR) of the data is 70 and the third and the first quartile of the data is (90, 20).

The second and third quartiles, or the middle half of your data collection, are contained in the interquartile range (IQR). The interquartile range provides the range of the middle half of a data set, whereas the range provides the spread of the entire data collection.

The given plot shows the number of days on the market for single-family homes in the city.

The figure shows the first quartile = 20 and the third quartile 90

Therefore the interquartile range of the data is equal to the difference between the third quartile and the first quartile i.

The interquartile range = Q3 - Q1

                                       = 90 - 20

                                       = 70

The third and first quartiles of the data are (90, 20).

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Answer: A | 12² + b² = 37²

Step-by-step explanation:

The height can be found by using the Pythagoras Theorem
h² = b² + p²

Answer:

Step-by-step explanation:

18 tickets cost $10 and 52 tickets cost $12.

The value of a in the given parabola is a = 4/5.

A quadratic function's graph is a parabola. A parabola, according to Pascal, is a circle's projection. Galileo described the parabolic route that projectiles take when they fall under the influence of uniform gravity. Several bodily movements have a curvilinear course that has the form of a parabola. In mathematics, a parabola is any planar curve that is mirror-symmetrical and typically resembles a U shape.

We know that the parabola passes through (-3, 0) and (5, 0), so we can write:

0 = a(-3)² + b(-3) + c (equation 1)

0 = a(5)² + b(5) + c (equation 2)

We also know that the parabola passes through (1, -32), so we can write:

-32 = a(1)² + b(1) + c (equation 3)

Equating the equation 1 and 2 we have:

9a - 3b + c = 25a + 5b + c

16a + 2b = 0 (equation 4)

Now, equation i can be written as:

c = - 9a + 3b

Substituting in equation 3 we have:

-32 = a + b - 9a + 3b

-32 = -8a + 4b

-8 = -2a + b (equation 5)

b = -8 + 2a

Substitute the value of b in equation 4:

16a + 2(-8 + 2a) = 0

16a - 16 + 4a = =

20a = 16

a = 16/20 = 4/5

Hence, the value of a in the given parabola is a = 4/5.

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Answer:

y = -x² - 2x + 15

Step-by-step explanation:

We can see that upto the x-value of -1, the y-value increases, and after -1, the y-value decreases. This tells us both that (-1, 16) is the vertex and that the function opens downwards.

The typical vertex form for quadratic equations is:

y= a(x-h)² + k where (h,k) is the vertex.

Replace the vertex (-1,16) for (h,k)

y = a (x+1)² +16

To find a, replace any value from the table. Let's use (3,0).

0 = a (3+1)² +16

0 = a (4)² + 16

0 = 16a + 16

-16 = 16a

a = -1

Now, insert that a into our  equation...

y = -13/16 (x+1)² +16

Then simplify:

y = -1 (x² +2x +1) + 16

y = -x² -2x -1 +16

We get our answer:

y = -x² - 2x + 15

Required factor form is 2x³(3x + 4).

What is GCF?

GCF stands for Greatest Common Factor, also known as the Greatest Common Divisor (GCD). In mathematics, the GCF of two or more numbers is the largest positive integer that divides each of the numbers without a remainder. For example, the GCF of 12 and 18 is 6 because 6 is the largest integer that divides both 12 and 18 without a remainder. The concept of GCF is important in many areas of mathematics, including algebra, number theory, and calculus, and is used in various problem-solving applications.

Given form is 6x⁴+8x³.

Here is two term 6x⁴ and 8x³.

By factorisation,

6x⁴ = 2×3×x⁴ and 8x³ = 2×2×2×x³

The greatest common factor (GCF) of 6x⁴ and 8x³ is 2x³.

To factor it out, we can divide each term by 2x³:

Now,

6x⁴ ÷ 2x³ = 3x

8x³ ÷ 2x³ = 4

So, we can write:

6x⁴ + 8x³ = 2x³(3x + 4)

Therefore, the factored form of 6x⁴ + 8x³ is 2x³(3x + 4).

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The angle of depression of the boat from the top of the cliff = 51.78 degrees

To calculate the angle of depression of the boat from the top of the cliff, we need to draw a diagram to visualize the situation.

In this instance, we can draw a right-angled triangle with one side representing the cliff's height (94m) and the other representing the horizontal distance between the boat and the cliff's foot. (74m). This triangle's hypotenuse depicts the line of sight from the top of the cliff to the boat.

The angle of depression is defined as the angle formed by the horizontal line and the line of sight from the cliff's summit to the boat.

We can use the tan function to determine this angle.

tan(angle of depression) = (height of cliff) / (horizontal distance from the boat to the foot of the cliff)

tan(∅) = (h/x)

tan(angle of depression) = 94 / 74

angle of depression = [tex]tanx^{-1} (94/74)[/tex]

angle of depression = 51.788974574 degrees

Therefore, the angle of depression of the boat from the top of the cliff is approximately 51.78 degrees.

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The angle of depression of the boat from the top of the cliff is approximately 50.67 degrees.

To calculate the angle of depression of the boat from the top of the cliff, we can use trigonometry. The angle of depression is the angle formed between the horizontal line (parallel to the ground) and the line of sight from the top of the cliff to the boat.

In this scenario, we have a right triangle formed by the cliff, the boat, and the horizontal line connecting them.

Let's denote the angle of depression as θ.

Using trigonometric ratios, we can use the tangent function to calculate the angle of depression:

tan(θ) = opposite/adjacent

In this case, the opposite side is the height of the cliff (94 meters) and the adjacent side is the horizontal distance from the foot of the cliff to the boat (74 meters).

tan(θ) = 94/74

To find θ, we can take the inverse tangent (arctan) of both sides:

θ = arctan(94/74)

Using a calculator or trigonometric table, we can find that θ is approximately 50.67 degrees.

Therefore, the angle of depression of the boat from the top of the cliff is approximately 50.67 degrees.

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Charlie must earn at least 125 tickets in the last game tο win the big prize.

A linear inequality is a mathematical statement that compares twο expressiοns using the symbοls < (less than), > (greater than), ≤ (less than οr equal tο), οr ≥ (greater than οr equal tο), and where bοth expressiοns are linear functiοns οf the same variables. It describes range οf values that the variables can take while satisfying the inequality.

Let x be the number οf tickets that Charlie must earn in the last game tο win the big prize.

The tοtal number οf tickets Charlie has won so far is the sum οf the tickets frοm all the games he has played:

95 + 115 + 90 + 75 = 375.

Tο win the big prize, Charlie must earn at least 500 tickets in tοtal.

Therefοre, we can write the fοllοwing inequality tο sοlve fοr the pοssible number οf tickets, x:

375 + x ≥ 500

Simplifying the inequality:

x ≥ 500 - 375

x ≥ 125

Therefore, Charlie must earn at least 125 tickets in the last game to win the big prize.

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Answer: 16a^2−24a−7

Step-by-step explanation:

The water level is rising at a rate of 0.0075 ft/min when the depth at the deepest point is 5 ft.

Let's call the depth at the deepest point of the pool "y" and the volume of

the water in the pool "V".

We want to find the rate at which the water level is rising, which is the

rate of change of "y" with respect to time.

We know the rate at which the pool is being filled, which is 0.9 ft3/min,

and we can find the rate at which the volume of the water is increasing

using the formula for the volume of a rectangular prism:

V = lwh

where l is the length, w is the width, and h is the height.

Since the pool is not rectangular, we can find the volume of the water as

a function of "y" using similar triangles:

h/y = (9 - 3)/(40 - 20) = 0.3

where h is the height of the pool at the deepest point. Solving for h, we get:

h = 0.3y

Substituting this expression for h into the formula for the volume, we get:

V = lw(3 + 0.6y)

Taking the derivative with respect to time, we get:

dV/dt = lw(0.6 dy/dt)

Now we need to find the values of l and w. The width is given as 20 ft,

and the length is a function of "y":

l = 40 - 2(40 - y) = 2y

Substituting these values into the expression for dV/dt, we get:

dV/dt = 40y(0.6 dy/dt)

Finally, we can find the rate at which the water level is rising when y = 5 ft

by plugging in the values:

0.9 = 40(5)(0.6 dy/dt)

Solving for dy/dt, we get:

dy/dt = 0.9/(4050.6) = 0.0075 ft/min

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From midnight to 5:00 am, the temperature dropped for 5 hours at 0.5°C each hour, so the total temperature drop was:

5 x 0.5°C = 2.5°C

Let the temperature at midnight be T°C. Then, we can use the information given to set up an equation:

Temperature at 5:00 am = Temperature at midnight - total temperature drop

11.5°C = T°C - 2.5°C

Solving for T, we can add 2.5°C to both sides of the equation:

11.5°C + 2.5°C = T°C - 2.5°C + 2.5°C

14°C = T°C

Therefore, the temperature at midnight was 14°C.

The integral of  probability density function [tex]\int_{4}^{\infty}[/tex]f(x) dx represents option b. the probability that a tree has a diameter of at least 4 inches.

Probability density function represented by function f.

Probability density function f for the diameter of trees in a forest is equals to ,

[tex]\int_{4}^{\infty}[/tex]f(x) dx

Because the integral is computing the area under the PDF curve for diameters greater than or equal to 4 inches.

And the area under a PDF curve represents the probability of the random variable in this case, tree diameter falling within that range.

This implies,

Integrating the PDF from 4 to infinity gives the probability of a tree having a diameter greater than or equal to 4 inches.

Therefore, the correct answer to represents the probability density function is Option (b). the probability that a tree has a diameter of at least 4 inches.

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The above question is incomplete , the complete question is :

Let f be the probability density function (PDF) for the diameter of trees in a forest, measured in inches. What does [tex]\int_{4}^{\infty}[/tex] f(x) dx represent?

(a) The standard deviation of the diameter of the trees in the forest.

(b) The probability that a tree has a diameter of at least 4 inches

(c) The probability that a tree has diameter less than 4 inches.

(d) The mean diameter of the trees in the forest.

The increasing levels of carbon dioxide in the atmosphere are causing significant changes to the hydrosphere, with far-reaching impacts on both marine and freshwater ecosystems.

What is Carbon dioxide ?

Carbon dioxide (CO2) is a gas that is naturally present in the Earth's atmosphere, and it plays a vital role in regulating the Earth's climate.

The increasing levels of carbon dioxide in the atmosphere are having a significant impact on the hydrosphere, which includes all the Earth's water systems such as oceans, lakes, rivers, groundwater, and ice caps.

When carbon dioxide dissolves in water, it reacts with it to form carbonic acid, which can cause the pH of the water to decrease, making it more acidic. This process is known as ocean acidification. As carbon dioxide levels continue to rise, the acidity of seawater is also increasing, which can have negative impacts on marine organisms that depend on certain pH levels to survive. For example, ocean acidification can interfere with the ability of shell-forming organisms such as corals, mollusks, and some plankton to build their shells, which can lead to population declines and changes in the food web.

In addition to ocean acidification, increased levels of carbon dioxide in the atmosphere can also contribute to rising sea levels due to melting ice caps and glaciers. The warming caused by increased greenhouse gases can also cause changes in precipitation patterns, leading to more frequent and severe floods and droughts that can impact freshwater resources.

Therefore,  the increasing levels of carbon dioxide in the atmosphere are causing significant changes to the hydrosphere, with far-reaching impacts on both marine and freshwater ecosystems.

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the possible whole number measures of the third side are 5, 6, and 7, listed in ascending order.

what is ascending order?

Ascending order refers to a sorting arrangement in which items are arranged from smallest to largest or from lowest to highest. For example, if you have a list of numbers such as 2, 7, 1, 9, 5, arranging them in ascending order would give you the sequence

In the given question,

According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. Therefore, in this case, we have:

6 + 2 > x

Where x is the length of the third side. Simplifying this inequality, we get:

8 > x

So, the third side must be less than 8.

Now, we also know that the third side must be greater than the difference between the first two sides:

6 - 2 < x

4 < x

So, the third side must be greater than 4.

Therefore, the possible whole number measures of the third side are 5, 6, and 7, listed in ascending order.

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Based on the given information and the secant-Secant Theorem, we have derived the equation (XY * XZ) = (10X * 10X).

Given that Circle U has points X and Z on the circle, and secant segments XY and ZY intersect at point Y outside the circle, we will use the following terms in our answer: Circle, Secant Segments, Intersection, and Ratio.

Step 1: Identify the segments involved.
We have two secant segments intersecting at point Y: XY and ZY.

Step 2: Apply the secant-secant theorem.
According to the secant-secant theorem, the product of the lengths of the segments from the intersection point to the points on the circle should be equal. In other words, (XY * XZ) = (ZY * ZY).

Step 3: Apply the given ratio.
We are given that ZY = 10X. Let's substitute this into the equation from Step 2: (XY * XZ) = (10X * 10X).

Step 4: Solve for XY and XZ.
Now, we need to solve for XY and XZ using the equation from Step 3. However, since we only have one equation and two variables, we cannot solve it completely. We need more information to find the specific values of XY and XZ.

Based on the given information and the secant-secant theorem, we have derived the equation (XY * XZ) = (10X * 10X). To find the specific values of XY and XZ, additional information is required.

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Answer:

24 = 31.4n + (–8.4)

Step-by-step explanation:

24=31.4n-8.4

Answer:

Step-by-step explanation:

ER=(5) [A single,1-tuple]. You start with 30 eggs. After selling 5 eggs, cooking 10, and frying 10 eggs 5 eggs remain.

PREMISES

ER=(E-S)-(C+F)

ASSUMPTIONS

Let E=the number of eggs at the start (30)

Let S=the number of eggs that were sold (5)

Let C=the number of eggs that were “cooked” i.e., hard-boiled, poached, and so forth (10)

Let F=the number of eggs that were fried (10)

Let ER=the number of eggs that remain having not been sold, cooked, or fried

CALCULATIONS

ER=(E-S)-(10+10)

ER=(30–5)-(20)

ER=25–20

ER=

5 eggs

PROOF

If ER=5, then the mathematic sentence ER=(E-S)-(C+F) returns

ER=(30–5)-(C+F)

5=25-(10+10)

5=25–20 and

5=5 verifies the result ER=5 of the sentence

Ike and Gus ran for 12 miles and 36 miles respectively.

Let us suppose, the number of miles Ike ran be x.

According to question, Gus ran 3 times as far as Ike.

So, Gus's running distance = 3x.

Also, Gus's and Ike's combined running distance = 48

⇒ x + 3x = 48

⇒ 4x = 48

⇒ x = 48 ÷ 4

⇒ x = 12.

∴ Ike ran 12 miles.

and distance covered by Gus = 3x

                                                 = 3 × 12

                                                 = 36.

Hence, Ike and Gus ran for 12 miles and 36 miles respectively.

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Rounding to four decimal places, the probability is (a) 0.3273 (b) 0.8423, and (c) 0.0446.

We can model this problem using a binomial distribution, where the probability of success (having the disease) is p = 0.84, and the number of trials (people tested) is n = 10.

(a) To find the probability that all 10 people have the disease, we can calculate the:

P(X = 10) = (10 choose 10) * (0.84)^10 * (1-0.84)^(10-10) = 0.3273

Rounding to four decimal places, the probability is 0.3273.

(b) To find the probability that at least 8 people have the disease, we can calculate the:

P(X ≥ 8) = P(X = 8) + P(X = 9) + P(X = 10)

We can find each term using the binomial probability formula and then add them up:

P(X = 8) = (10 choose 8) * (0.84)^8 * (1-0.84)^(10-8) = 0.2018

P(X = 9) = (10 choose 9) * (0.84)^9 * (1-0.84)^(10-9) = 0.3132

P(X = 10) = (10 choose 10) * (0.84)^10 * (1-0.84)^(10-10) = 0.3273

P(X ≥ 8) = 0.2018 + 0.3132 + 0.3273 = 0.8423

Rounding to four decimal places, the probability is 0.8423.

(c) To find the probability that at most 4 people have the disease, we can calculate:

[tex]P(X ≤ 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)[/tex]

We can find each term using the binomial probability formula and then add them up:

P(X = 0) = (10 choose 0) * (0.84)^0 * (1-0.84)^(10-0) = 0.000004

P(X = 1) = (10 choose 1) * (0.84)^1 * (1-0.84)^(10-1) = 0.0001

P(X = 2) = (10 choose 2) * (0.84)^2 * (1-0.84)^(10-2) = 0.0012

P(X = 3) = (10 choose 3) * (0.84)^3 * (1-0.84)^(10-3) = 0.0084

P(X = 4) = (10 choose 4) * (0.84)^4 * (1-0.84)^(10-4) = 0.0348

P(X ≤ 4) = 0.000004 + 0.0001 + 0.0012 + 0.0084 + 0.0348 = 0.0446

Rounding to four decimal places, the probability is 0.0446.

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The graph of the linear equation y = mx + c is a line with m as the slope, and m, and c as the y-intercept. This form of the linear equation is called the slope-intercept form, and the values of m and c are real numbers.

The slope, m, represents the steepness of a line. The slope of the line is also termed a gradient, sometimes. The y-intercept, b, of a line, represents the y-coordinate of the point where the graph of the line intersects the y-axis.

Here, the distance c is called the y-intercept of the given line L.

So, the coordinate of a point where the line L meets the y-axis will be

(0, c). That means line L passes through a fixed point (0, c) with slope m.

We know that, the equation of a line in point-slope form, where (x1, y1) is the point and slope m is:

(y – y1) = m(x – x1)

Here, (x1, y1) = (0, c)

Substituting these values, we get;

y – c = m(x – 0)

y – c = mx

y = mx + c

Therefore, the point (x, y) on the line with slope m and y-intercept c lies on the line if and only if y = mx + c

Note: The value of c can be positive or negative based on the intercept made on the positive or negative side of the y-axis, respectively

Slope Intercept Form x Intercept

We can write the formula for the slope-intercept form of the equation of line L whose slope is m and x-intercept d as:

y = m(x – d)

Here,

m = Slope of the line

d = x-intercept of the line

Sometimes, the slope of a line may be expressed in terms of tangent angle such as:

m = tan θ

Answer:

its better to buy the cylinder because it can hold more

Step-by-step explanation:

Without specific dimensions for the containers, we cannot provide exact values for volume and unit rate.

To determine which popcorn container has a better unit rate and holds more popcorn, we first need to find the volume of both containers.

Assuming the given dimensions are radius and height, let's say the cone has a radius of 'r1' and a height of 'h1,' and the cylinder has a radius of 'r2' and a height of 'h2.'

The volume of a cone (V1) is given by the formula: V1 = (1/3)πr1²h1
The volume of a cylinder (V2) is given by the formula: V2 = πr2²h2

Next, we need to find the unit rate of both containers. Divide the volume of each container by its price.

Unit rate of cone: UR1 = V1 / $6.75
Unit rate of cylinder: UR2 = V2 / $6.25

To determine which container holds the most popcorn, compare V1 and V2. The container with the larger volume holds more popcorn.

To determine which container has a better unit rate, compare UR1 and UR2. The container with the higher unit rate is a better buy per $1.

However, you can plug in the given dimensions for your problem and follow these steps to find your answer.

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1331.046

Step-by-step explanation:

Thus, the number of coin for each are- number of dimes be 5 and number of quarters be 1.

A linear system can be algebraically solved using the substitution approach. One y-value is substituted for another in the substitution procedure. Getting the value of the x-variable in regards of the y-variable is the method's most straightforward step.

Let the number of dimes be 'x'.

Let the number of quarters be 'y'.

Then,

Total amount of parking meter is 6.25.

x + y = 6.25

x = 6.25  - y ....eq 1

Now,

number of dimes is 2 more than 3 times the number of quarters.

So,

x = 3y + 2  ...eq 2

Equating eq 1 and 2

6.25  - y = 3y + 2

4y = 4.25

y = 1.062

y = 1 (approx)

x = 3(1.065) + 2

x = 5.195

x = 5 (approx)

Thus, the number of coin for each are- number of dimes be 5 and number of quarters be 1.

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We may assume that out of the 300 graduating students, 120 plan to go

to college, 75 want to work, 60 are unsure, and 45 want to join the

military. It's critical to remember that these are estimates based on a poll

of 20 peers, and actual figures may vary slightly.

Based on the data from the survey of 20 classmates, we can make

inferences about the plans of the entire graduating class of 300 students

using proportions.

Let's fill in the table based on the given information:

Plan Number of classmates Proportion

College 8 8/20 = 0.4

Work 5 5/20 = 0.25

Undecided 4 4/20 = 0.2

Military 3 3/20 = 0.15

To find the number of students in each category out of 300, we can

multiply the proportion by 300:

College: 0.4 x 300 = 120 students

Work: 0.25 x 300 = 75 students

Undecided: 0.2 x 300 = 60 students

Military: 0.15 x 300 = 45 students

Therefore, we can infer that out of the 300 students in the graduating

class, approximately 120 plan to attend college, 75 plan to work, 60 are

undecided, and 45 plan to enter the military. It's important to note that

these are estimates based on the survey of 20 classmates, and there

may be some variability in the actual numbers.

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Zach's hourly wage is determined by taking the basic rate of $16.00 and adding $2.00 for each extra student beyond the first one in the calculation is P = $16.00 + ($2.00 x (n - 1))

That's an interesting situation! It seems like Zach's hourly pay for tutoring is not fixed, but rather increases with each additional student he tutors. Here's how his pay works:

If Zach is tutoring just one student, he gets paid $16.00 per hour.

If a second student joins the tutoring session, Zach's hourly pay increases to $18.00 per hour ($16.00 for the first student plus $2.00 for the second).

If a third student joins, Zach's hourly pay increases to $20.00 per hour ($16.00 for the first student plus $2.00 for the second and third students).

And so on, for each additional student who joins the tutoring session.

So Zach's hourly pay is not a fixed rate, but rather a function of the number of students he is tutoring.

If we let P be Zach's hourly pay and n be the number of students he is tutoring, we can express Zach's pay as:

P = $16.00 + ($2.00 x (n - 1))

This formula calculates Zach's hourly pay by taking the base rate of $16.00 and adding $2.00 for each additional student beyond the first one.

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Question:-  Zach got hired to tutor math. He gets paid $16.00 per hour for tutoring one student. Each student that joins the tutoring session increases his hourly pay by $2.00.

Answer:

Greatest is Miami

Least is Washington

Step-by-step explanation:

Population/Area= Population Density

Ex:

Population 2

Area 1

2/1=2

Answer:

  50 ft by 25 ft . . . . . 50 ft parallel to the barn

Step-by-step explanation:

You want the dimensions of the largest rectangular area that can be enclosed using 100 ft of fence for three sides.

If the dimensions of the space are L feet in length and W feet in width, where L is parallel to the barn, the length of the perimeter fence is ...

  P = L +2W

Solving for W gives ...

  W = (P -L)/2

The area of the enclosed space is ...

  A = LW

  A = L(P -L)/2 . . . . . substitute for W

The area formula is the equation for a parabola that opens downward. It has zeros at L=0 and at L=P. The vertex (maximum) is found at the value of L that lies on the line of symmetry, halfway between these zeros. If we call that length M, then we have ...

  M = (0 +P)/2 = P/2

The length of enclosure that maximizes the area is 1/2 the length of the available fence.

The width is ...

  W = (P -P/2)/2 = P/4

The width of the enclosure that maximizes the area is 1/4 the length of the available fence.

Using 100 feet of fence, the dimensions are ...

__

Additional comment

Note that we have solved this in a generic way. The solution given is the general solution to the 3-sided enclosure problem.

This is a special case of the general rectangular enclosure problem which has the solution that the total cost in one direction is equal to the total cost in the orthogonal direction. This rule applies even when costs are different for the different sides or for any partitions that might divide the enclosure.

Here the "cost" is simply the length of the fence. The 50 ft of fence parallel to the barn is equal in length to the 25 +25 ft of fence perpendicular to the barn.

The dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).

To maximize the area of the rectangular fence, follow these steps:

1. Let's assign variables to the dimensions: let the length of the rectangle parallel to the barn wall be x feet, and the length perpendicular to the barn wall be y feet.

2. We are given that 100 feet of fence will be used for the other three sides. This means the fencing equation is:

x + 2y = 100.

3. Solve for x:  x = 100 - 2y.

4. The area A of the rectangle is given by the product of its dimensions: A = xy.

5. Substitute the expression for x from step 3 into the area formula:  A = (100 - 2y)y.

6. Expand the expression: A = 100y - 2y^2.

7. To maximize the area, we need to find the maximum value of the quadratic function A(y). Since the coefficient of the y^2 term is negative, the graph of A(y) is a downward-opening parabola, which means it has a maximum value.

8. To find the maximum, we'll use the vertex formula for parabolas: y_vertex = -b/(2a), where a = -2 and b = 100. Plugging in these values, we get y_vertex = -100/(2 * -2) = 25.

9. Substitute the value of y_vertex back into the equation for x: x = 100 - 2(25) = 50.

10. So the dimensions that give the maximum area inside the rectangle are x = 50 feet (parallel to the barn wall) and y = 25 feet (perpendicular to the barn wall).

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