Find the unique function f(x) satisfying the following conditions: f" (x) = x2 f(1) 4 f(2) = 1 f(x) =

The total surface area of the foam pit can be calculated by finding the area of each face and adding them together.

Since the pit is a rectangular prism, it has six faces: the top, bottom, front, back, left, and right. The area of each face can be calculated using the formula for the area of a rectangle, which is length times width.

To calculate the total surface area of a rectangular prism with an open top, we need to add the areas of all six faces together.

The area of each face can be calculated using the formula for the area of a rectangle (length times width).

The top of the foam pit is open, so we don't need to include it in our calculation.

After finding the area of each face, we simply add them all together to get the total surface area.

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The probability that four boys are given the first four seats on the first teacup is approximately equal to 0.004.

To determine the probability that four boys are given the first four seats on the first teacup.

The total number of ways to distribute 12 tickets among 12 seats is 12! (12 factorial), which is equal to 479,001,600.

We need to find the number of ways that four boys can be selected from the seven boys, multiplied by the number of ways that eight people (including the remaining three boys and five girls) can be selected from the ten remaining people,

multiplied by the number of ways that the selected people can be arranged on the teacup ride.

The number of ways to select four boys from seven boys is 7C4, which is equal to 35. The number of ways to select eight people from the remaining ten people is 10C8, which is equal to 45.

Finally, the number of ways to arrange the selected twelve people on the teacup ride is 4!, which is equal to 24.

Therefore, the probability that four boys are given the first four seats on the first teacup is (35 x 45 x 24) / 12!, which is approximately equal to 0.004.

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The total number of gifts is given as follows:

439 gifts.

The Fundamental Counting Theorem states that if there are m ways to do one thing and n ways to do another, then there are m x n ways to do both.

This can be extended to more than two events, where the number of ways to do all the events is the product of the number of ways to do each individual event, according to the equation presented as follows:

[tex]N = n_1 \times n_2 \times \cdots \times n_n[/tex]

For a single gift, the number of options is given as follows:

10 + 4 + 7 = 21 gifts.

For two gifts, the number of options is given as follows:

10 x 4 + 10 x 7 + 7 x 4 = 138 gifts.

For three gifts, the number of options is given as follows:

10 x 4 x 7 = 280 gifts.

Hence the total number of gifts is obtained as follows:

280 + 138 + 21 = 439 gifts.

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[tex]\sf y =\dfrac{2}{5}x-4[/tex]

Step-by-step explanation:

        To find the equation of the required line, first we need to find the slope of the given line in the graph.

          Choose two points from the graph.

         (0 ,4)    x₁ = 0 & y₁ = 4

         (2,-1)     x₂ = 2 & y₂ = -1

[tex]\sf \boxed{\sf \bf Slope=\dfrac{y_2-y_1}{x_2-x_1}}[/tex]

           [tex]\sf = \dfrac{-1-4}{2-0}\\\\=\dfrac{-5}{2}[/tex]

       [tex]\sf m_1=\dfrac{-5}{2}[/tex]

[tex]\sf \text{Slope of the perpendicular line m = $\dfrac{-1}{m_1}$}[/tex]

                                                    [tex]\sf = -1 \div \dfrac{-5}{2}\\\\=-1 * \dfrac{-2}{5}\\\\=\dfrac{2}{5}[/tex]

[tex]\boxed{\sf slope \ intercept \ form \ : \ y = mx + b}[/tex]

Here, m is slope and b is y-intercept.

Substitute the m value in the above equation,

                    [tex]\sf y =\dfrac{2}{5}x + b[/tex]

    The line is passing through (5 , -2),

                     [tex]\sf -2 = \dfrac{2}{5}*5+b[/tex]

                     -2 =  2 + b

                -2 - 2 = b

                      b = -4

Slope-intercept form:

                [tex]\sf y = \dfrac{2}{5}x-4[/tex]

The closest amount to the total balance is option B, 695.00.

For Account A, the interest earned after 2 years is:

Interest = Principal * Rate * Time = 400 * 0.035 * 2 = 28

So the total balance in Account A after 2 years is:

Total A = Principal + Interest = 400 + 28 = 428

For Account B, the interest earned after 2 years is:

Interest = Principal * (1 + Rate/100)^Time - Principal = 250 * (1 + 3.25/100)^2 - 250 = 25.95

The total balance in Account B after 2 years is:

Total B = Principal + Interest = 250 + 25.95 = 275.95

The total balance in both accounts after 2 years is:

Total Balance = Total A + Total B = 428 + 275.95 = 703.95

The closest amount to the total balance is option B, 695.00.

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The vertex form of the equation of the parabola is:

r = (1/24)(y - 1)^2 + 3

Since the directrix is a horizontal line, the axis of the parabola is vertical. Therefore, the vertex form of the equation of the parabola is:

r = a(y - k)^2 + h

where (h, k) is the vertex of the parabola and "a" is a constant that determines the shape and orientation of the parabola.

Since the focus is (3,-5), the vertex of the parabola is halfway between the focus and the directrix. The directrix is 6 units above the vertex, so the vertex is (3,1).

We can use this information to write the vertex form of the equation:

r = a(y - 1)^2 + 3

To find the value of "a", we need to use the distance formula between the vertex and the focus:

distance = |y-coordinate of focus - y-coordinate of vertex| = 6

| -5 - 1 | = |-6| = 6

Using the definition of the parabola, the distance from the vertex to the focus is also equal to 1/(4a). Therefore:

1/(4a) = 6

a = 1/(4*6) = 1/24

Substituting this value of "a" into the vertex form equation, we get:

r = (1/24)(y - 1)^2 + 3

Therefore, the vertex form of the equation of the parabola is:

r = (1/24)(y - 1)^2 + 3

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After formula for the volume of a cylinder, the increase in water level results in approximately 260 more cubic feet of water in the tank.

The current water level is 10 feet, which is 120 inches. When the water level increases by 2 inches, the new water level will be 122 inches.

The radius of the tank is half of the diameter, which is 30 feet or 360 inches.

The current volume of water in the tank can be calculated using the formula for the volume of a cylinder: V = πr²h, where r is the radius and h is the height of the water level.

V = π(360²)(120) ≈ 15,465,920 cubic inches

When the water level increases by 2 inches, the new height of the water level is 122 inches.

The new volume of water in the tank can be calculated using the same formula:

V = π(360²)(122) ≈ 15,914,693 cubic inches

The difference in volume between the two levels is:

15,914,693 - 15,465,920 = 448,773 cubic inches

To convert cubic inches to cubic feet, we divide by 1728:

448,773 ÷ 1728 ≈ 259.6 cubic feet

Rounding to the nearest cubic foot, we get:

260 cubic feet

Therefore, the increase in water level results in approximately 260 more cubic feet of water in the tank.

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A random sample that has 1000 college students in the United States is surveyed about how much money they spend on books per year, and the mean amount calculated is 1000 college students in the US. Option A is the correct answer.

The sample in this scenario refers to the group of college students who were surveyed about their book spending habits. In this case, the sample size is 1000 college students in the United States.

The purpose of this survey is to estimate the mean amount of money spent on books per year by college students in the US, using the sample mean as an estimate. It is important to note that the sample should be representative of the larger population of college students in the US.

Therefore, option A, "1000 college students in the US," is the correct answer. Option B, "all college students in the US," represents the population, not the sample. Options C and D are not relevant to the given scenario.

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The question is -

A random sample of 1000 college students in the United States is surveyed about how much money they spend on books per year, and the mean amount is calculated. What is the sample?

a. 1000 college students in the US

b. all college students in the US

c. 1000 college students in CA

d. all college students in CA

√6a + 3 < a + 2,  we have proven that √6a + 3 < a + 2 for all a > 1 using the Mean Value Theorem.

To use the Mean Value Theorem to prove √6a + 3 < a + 2 for all a > 1, we first note that the function f(x) = √6x + 3 is continuous on the interval [1,a].

Next, we need to find a point c in the interval (1,a) such that the slope of the line connecting (1,f(1)) and (a,f(a)) is equal to the slope of the tangent line to f(x) at c.

The slope of the line connecting (1,f(1)) and (a,f(a)) is given by:

(f(a) - f(1)) / (a - 1) = (√6a + 3 - √9) / (a - 1) = (√6a) / (a - 1)

To find the slope of the tangent line to f(x) at c, we first find the derivative of f(x):

f'(x) = (1/2) * (6x + 3)^(-1/2) * 6 = 3 / √(6x + 3)

Then, we evaluate f'(c) to get the slope of the tangent line at c:

f'(c) = 3 / √(6c + 3)

Now, by the Mean Value Theorem, there exists a point c in (1,a) such that:

f'(c) = (√6a) / (a - 1)

Setting these two expressions for f'(c) equal to each other, we get:

3 / √(6c + 3) = (√6a) / (a - 1)

Solving for c, we get:

c = (a + 2) / 6

(Note that c is indeed in (1,a) since a > 1.)

Now, we can evaluate f(a) and f(1) and use the Mean Value Theorem to show that:

√6a + 3 - √9 < (√6a) / (a - 1) * (a - 1)

Simplifying, we get:

√6a + 3 < a + 2

as desired.

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By compound interest, The initial amount in the account is $300.

What does compound interest mean ?

When you earn interest on your interest earnings as well as the money you have saved, this is known as compound interest. As an illustration, if you put $1,000 in an account that offers 1% yearly interest, you will receive $10 in interest after a year.

                             Compound interest allows you to earn 1 percent on $1,010 in Year Two, which equates to $10.10 in interest payments for the year. This is possible because interest is added to the principle in Year Two.

A = 300(1.035)t

As we know the formula "Compound Interest" :

A = P(1 + r/100)t

So, According to our question,

Rate of interest = 0.35 = 135%

So, equate the both the equations , we get that

Hence, The initial amount in the account = $300

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The student from the first class performs better when comparing their scores using z-scores.

To compare the students' performances, we will calculate their z-scores, which show how many standard deviations away their scores are from the mean of their respective classes.

For the student from the first class:
z-score = (Score - Mean) / Standard Deviation
z-score = (62 - 56) / 9
z-score ≈ 0.67

For the student from the second class:
z-score = (83 - 75) / 15
z-score ≈ 0.53

The student from the first class has a higher z-score (0.67) compared to the student from the second class (0.53). This means the student from the first class performed better relative to their classmates.

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Part A:

The definition provided for a circle is actually correct. However, if we change the definition slightly to say that a circle is the set of all points in a plane that are the same distance from a given point, we can find an example that contradicts it.

For instance, consider a cone in three-dimensional space. If we take a cross-section of the cone that is parallel to the base, we get a circle. However, this circle is not the set of all points that are the same distance from one given point, but rather from the axis of the cone.

To make the definition more accurate, we need to specify that the circle exists in a plane.

Part B:

An example of an undefined term related to a circle is the term "point." A circle is defined as the set of all points that are the same distance from a given point, but the term "point" is not defined within this definition.

A point is typically defined as a location in space that has no size or shape. In the context of a circle, a point can be thought of as any location on the circumference of the circle. However, it is important to note that the definition of a point is not dependent on the definition of a circle, and vice versa.

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The series 2n^2 is divergent.

To express the nth partial sum as a telescoping sum, we need to find a pattern in the terms of the series.

The general term of the series is given by an = 2n^2 - ?.

The nth partial sum can be written as:

sn = a1 + a2 + a3 + ... + an

= 2(1)^2 - ? + 2(2)^2 - ? + 2(3)^2 - ? + ... + 2n^2 - ?

We can simplify the above expression by factoring out 2 from each term:

sn = 2(1^2 + 2^2 + 3^2 + ... + n^2) - n?

Using the formula for the sum of squares, we have:

sn = 2(n(n+1)(2n+1)/6) - n?

Simplifying further, we get:

sn = (n^3 + 3n^2 + 2n)/3 - n?

Taking the limit as n approaches infinity, we get:

lim n->∞ sn = lim n->∞ [(n^3 + 3n^2 + 2n)/3 - n?]

Since the term n? grows without bound as n approaches infinity, the limit of sn does not exist.

Therefore, the series 2n^2 - ? is divergent.

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Answer:

part a: The slope of the function represents the rate of change of the balance in the bank account per day. To find the slope, we can use the formula: slope = (change in y)/(change in x).

Using the values from the table, we have: slope = (720-600)/(3-0) = 120/3 = 40. Therefore, the slope of the function g(x) is 40.

part b: Using the point-slope form of the equation of a line, we can write: g(x) - 600 = 40(x-0). Simplifying, we get: g(x) = 40x + 600. This is the slope-intercept form of the equation, where the y-intercept is 600 and the slope is 40.

To write in standard form, we can rearrange the equation as: -40x + g(x) = 600.

part c: Using function notation, we can write the equation as: g(x) = 40x + 600.

part d: To find the balance in the bank account after 7 days, we can use the equation we found in part c and substitute x = 7: g(7) = 40(7) + 600 = 880. Therefore, the balance in the bank account after 7 days is $880.

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Dominick has earned $6,465.55 - $4,420.00 = $2,045.55 more interest than Ryan after 10 years.

To solve this problem, we can use the formulas for compound interest and simple interest.

Compound interest formula:

[tex]A = P(1 + r/n)^(^n^t^)[/tex]

Where:

A = the amount after time t

P = the principal

r = the annual interest rate

n = the number of times the interest is compounded per year

t = time in years

Simple interest formula:

I = Prt

Where:

I = the interest earned

P = the principal

r = the annual interest rate

t = time in years

Using the compound interest formula for Dominick's account:

[tex]A = P(1 + r/n)^(^n^t^)[/tex]

A = 6500(1 + 0.068/365)^(365*10)

A ≈ $12,965.55

Using the simple interest formula for Ryan's account:

I = Prt

I = 65000.06810

I = $4,420.00

Dominick's account has earned: $12,965.55 - $6,500 = $6,465.55 in interest.

Ryan's account has earned: $4,420.00 in interest.

Therefore, Dominick has earned $6,465.55 - $4,420.00 = $2,045.55 more interest than Ryan after 10 years.

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(Explanation below)

x=2

x = 2 is the solution of the equation

Two or more expressions with an Equal sign is called as Equation.

The given equation is [tex](3X-5)^(^1^/^4^) + 3 = 4[/tex]

We have to find the value of x

Subtracting 3 from both sides:

[tex](3X-5)^(^1^/^4^) = 1[/tex]

Raising both sides to the fourth power:

3X - 5 = 1^4

3X - 5 = 1

Adding 5 to both sides:

3X = 6

Dividing by 3:

X = 2

Therefore, x = 2 is the solution of the equation

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The relation a into b represent  all possible combinations of elements in a and b as ordered pairs.

The relation "a into b" refers to the cartesian product of the sets a and b, which is denoted by a × b. The cartesian product of two sets is a set of all possible ordered pairs, where the first element of each ordered pair comes from the first set, and the second element comes from the second set.

In this case, a = {0, 1, 2} and b = {-1, 1, 2}. So, their cartesian product a × b is the set of all possible ordered pairs (a, b), where a is an element of a and b is an element of b. Therefore, we have:

a × b = {(0, -1), (0, 1), (0, 2), (1, -1), (1, 1), (1, 2), (2, -1), (2, 1), (2, 2)}

This means that the relation a into b consists of all possible combinations of elements in a and b as ordered pairs.

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Answer:

log base 4 (5x + 4) = 3

Step-by-step explanation:

a^b=c  is  loga(c)=b

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The completed table with residuals rounded to hundredths place:

| Hours | Retweets | Predicted Value | Residual |

| 1     | 65       |-79.86          |-144.86   |

| 2     |90        |-58.96          |-31.04    |

|3      |162       |-20.16          |-141.84   |

|4      |224       |17.64           |-206.64   |

|5      |337       |75.54           |-262.54   |

|6      |466       |133.44          |-332.44   |

|7      |780       |191.34          |-409.34   |

|8      |1087      |249.24          |-238.24   |

We can evaluate the predicted value by staging the given hours in the function

f(x) = 138.9x - 218.76.

for instance,  hours = 1:

f(1) = (138.9 x 1) - 218.76

= -79.86

likewise, we can find predicted values for all hours.

Residual = Actual Value - Predicted Value

For instance, for hours = 1:

Residual = Actual Value - Predicted Value

       = 65 - (-79.86)

       = 144.86

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After 16 years, compounded yearly, Lizzie will have $5,995.68 for college in Option A and $5,789.95 in Option B, rounded to the nearest dollar.

For Option A, the interest rate is 3% compounded yearly. Using the formula for compound interest, we have:

FV = PV x (1 + r)ⁿ

where FV is the future value, PV is the present value, r is the annual interest rate as a decimal, and n is the number of years. Plugging in the values, we get:

FV = $3000 x (1 + 0.03)¹⁶ = $5,995.68

For Option B, the interest rate is 2% compounded quarterly. We need to convert the annual interest rate to a quarterly rate by dividing it by 4. Then we use the formula:

FV = PV x (1 + r/n)^(n x t)

where r/n is the quarterly interest rate, n is the number of compounding periods per year, and t is the total number of years. Plugging in the values, we get:

FV = $3000 x (1 + 0.02/4)^(4 x 16) = $5,789.95

Therefore, Lizzie will have $5,995.68 for college in Option A and $5,789.95 in Option B, rounded to the nearest dollar.

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All interior angles are equal, so Angle ABE = 120°.
the exterior angle is equal to 60 degrees.

Angle BCE is equal to 180 degrees.

The polygon must have 18 sides because its exterior angles sum to 360°, and each exterior angle is 20°.

1) In a regular hexagon:
a) Angle ABE is an interior angle. To calculate the size of Angle ABE, we first find the sum of interior angles of a hexagon, which is (n-2)×180°, where n is the number of sides.

For a hexagon, n = 6, so the sum of interior angles is (6-2)×180° = 720°. Since it's a regular hexagon, all interior angles are equal, so Angle ABE = 720°/6 = 120°.

b) Angle DCE is an exterior angle. In a regular hexagon, the exterior angles are equal. To find the size of an exterior angle, we can use the formula: exterior angle = 360°/n, where n is the number of sides. For a hexagon, n = 6, so Angle DCE = 360°/6 = 60°.

c) Angle BEC is the sum of Angle ABE and Angle DCE. Therefore, Angle BEC = 120° + 60° = 180°.

2) For a regular polygon with an exterior angle of 20°:
a) The sum of the interior angle and exterior angle for any polygon is 180°. So, the size of its interior angle = 180° - 20° = 160°.

b) To find the number of sides in the polygon, we can use the formula for the exterior angle: exterior angle = 360°/n, where n is the number of sides. We know that the exterior angle is 20°, so 20° = 360°/n.

Solving for n, we get n = 360°/20° = 18 sides. The polygon must have 18 sides because its exterior angles sum to 360°, and each exterior angle is 20°.

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The exponential term (87^t)^(1.75) increases, leading to an exponential growth in the number of people who receive the email.

The relationship between the elapsed time t, in days, since Bharat sent the letter and the number of people P(t) who receive the email is modeled by the following function:

P(t) = 2401 * (87^t)^(1.75)

In this function, t represents the number of days that have passed since Bharat sent the letter, and P(t) represents the number of people who receive the email at that time.

The function is an exponential growth model where the base is 87, and the exponent is t raised to the power of 1.75. The constant 2401 is a scaling factor that determines the initial number of people who receive the email at t=0.

As time passes, the exponential term (87^t)^(1.75) increases, leading to an exponential growth in the number of people who receive the email.

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If we are drawing without replacement, the probability is approximately 27.7%, which is closest to option B: 27%.

The probability of drawing a yellow marble on the next draw depends on whether we are drawing with or without replacement.

If we are drawing without replacement, then the probability of drawing a yellow marble is 13 out of the remaining 47 marbles, since we have already drawn 35 white marbles and 13 yellow marbles out of the 48 total marbles.

If we are drawing with replacement, then the probability of drawing a yellow marble on the next draw is still 1/3, or approximately 33.3%.

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The equation y = (4/5)x - 3 is in slope-intercept form, y = mx + b, where m is the slope of the line. Therefore, the slope of the line represented by the equation is:

m = 4/5

So the answer is C) 4/5.

Answer:

Mean=15.86

Median=18

Mode=18

The value of m∠MGA between the tangent and the diameter is 90 degrees.

A line that touches the circle at one point is known as a tangent to a circle.

Therefore, MU is tangent to the circle O at the point G. The diameter of the

circle is GA.

Therefore, let's find the angle m∠MGA in the circle.

If a tangent and a diameter meet at the point of tangency, then they are perpendicular to one another.  Therefore, the point where they meets form a right angle(90 degrees).

Hence,

m∠MGA  = 90 degrees

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If Kayla can pay off up to $175 monthly for the purchase of a new television for $487.89, the interest she will pay at 23.5% compounded monthly is D. $18.87.

The compound interest payable on the credit card for the purchase of a new television can be computed using an online finance calculator as follows:

N (# of periods) = 3 months

I/Y (Interest per year) = 23.5%

PV (Present Value) = $487.89

PMT (Periodic Payment) = $-175

Results:

FV = $18.23

Sum of all periodic payments = $525.00

Total Interest = $18.87

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Answer:

6, 9, 10, 10, 11, 11, 12, 13, 15, 15, 18

Q1 (the first quartile) represents the data point that separates the lowest 25% of the data from the rest of the data. To find Q1, we can use the formula:

Q1 = (n + 1) / 4

where n is the total number of data points.

In this case, n = 11, so:

Q1 = (11 + 1) / 4 = 3rd data point

So, Q1 is 10.

Q3 (the third quartile) represents the data point that separates the highest 25% of the data from the rest of the data. To find Q3, we can use the formula:

Q3 = 3(n + 1) / 4

In this case:

Q3 = 3(11 + 1) / 4 = 9th data point

So, Q3 is 15.

D4 represents the fourth decile, which is the data point that separates the lowest 40% of the data from the rest of the data. To find D4, we can use the formula:

D4 = (n + 1) / 10 * 4

In this case:

D4 = (11 + 1) / 10 * 4 = 5th data point

So, D4 is 11.

D Assimilation represents the data point that is closest to the mean (average) of the data. To find D Assimilation, we first need to find the mean of the data:

Mean = (6 + 9 + 10 + 10 + 11 + 11 + 12 + 13 + 15 + 15 + 18) / 11 = 12

The data point closest to the mean is 12, so:

D Assimilation = 12

Pss (the range) represents the difference between the largest and smallest data points. In this case:

Pss = 18 - 6 = 12
6  9  10 10 11 11 12 13 15 15 18

                             Dss=12

   Q1=10       Q3=15

       D4=11

Step-by-step explanation:

Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

The rule of 5% slack states that the cable should have 5% of the cable's length in slack, so for a 130 feet cable, there would need to be 6.5 feet of slack. Since Chantal's design only has 10 feet of vertical change, that would not leave enough slack to meet the rule.

Also, the 6-8 feet of vertical change per 100 feet of horizontal change rule states that the vertical change should be between 6-8 feet for every 100 feet of horizontal change. In this case, the horizontal change is 130 feet, so the vertical change should be between 7.8-10.4 feet. Since Chantal's design has only 10 feet of vertical change, it does not meet this rule either.

Therefore, Chantal's design will not work as it does not meet the length requirements for the desired amount of slack and vertical change.

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you will have to walk approximately 17.2 miles back home.

A right triangle is a triangle in which one of the angles measures 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse, and the other two sides are called the legs.  the length of the hypotenuse. Right triangles have many important applications in mathematics, science, and engineering, particularly in trigonometry, which is the study of the relationships between the sides and angles of triangles.

You have formed a right triangle with legs of length 10 and 14. To find the length of the hypotenuse, which is the distance back to your starting point, we can use the Pythagorean theorem:

[tex]c^2 = a^2 + b^2[/tex]

[tex]c^2 = 10^2 + 14^2[/tex]

[tex]c^2 = 100 + 196[/tex]

[tex]c^2 = 296[/tex]

c = sqrt(296)

c ≈ 17.2

Therefore, you will have to walk approximately 17.2 miles back home.

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