Libby starts draining the pool for cleaning. The function y = 10,080 - 720x represents the


gallons of water y remaining in the pool after x hours. Find the zero and explain what it means in the context of the situation

Based on the following  observations, we can write the equation of the rational function as: f(x) = (x + 1)/(x - 1)

A rational function is a type of mathematical function that is defined as the ratio of two polynomial functions.

In other words, it is a function that can be expressed as f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) is not the zero polynomial.

To find the equation of the rational function represented by the given graph, we need to analyze the behavior of the graph and identify its key features. given below are the steps:

Look at the behavior of graph as x approaches infinity and negative infinity. The graph appears to have horizontal asymptotes at y = -1 and y = 1. This suggests that the function has a degree of 1 in both the numerator and denominator.

Identify any vertical asymptotes. The graph have vertical asymptote at x = 1. This suggests that the denominator of the function has a factor of (x - 1).

Look for any x-intercepts or y-intercepts.The graph's x-intercept and y-intercept are both at x = -1 and 1, respectively. This suggests that the numerator of the function has a factor of (x + 1) and that the function has a constant term of 1 in the numerator.

This function has a degree of 1 in both the numerator and denominator, a vertical asymptote at x = 1, and horizontal asymptotes at y = -1 and y = 1. It also has an x-intercept at x = -1 and a y-intercept at y = 1, which match the features of the graph given.

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The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

What are inequalities ?

Inequality is a relation that compares two numbers or other mathematical expressions in an unequal way.

here , we have,

The symbol a < b indicates that a is smaller than b.

When a > b is used, it indicates that a is bigger than b.

a is less than or equal to b when a notation like a ≤ b.

a is bigger or equal value of an is indicated by the notation a ≥ b.

Assuming the cost of soccer cleats to be 'C' and the number of subscriptions to be 's'.

∴ The inequality that represents this situation is 3s + 25 ≥ C to real his goal.

Hence, The inequality to describe the number of subscriptions Andre must sell to reach his goal is  3s + 25 ≥ C.

The required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.

A triangle with two sides that are perpendicular to one another is known as a right triangle, right-angled triangle, or orthogonal triangle. It was previously known as a rectangled triangle. Trigonometry is based on the relationship between the right triangle's sides and other angles.

Given data

GH = 14 units and GJ = 16 units

Using Pythagorean theorem;

[tex]16^2 = 14^2 + HJ^2[/tex]

[tex]HJ = \sqrt{16^2-14^2}[/tex]

HJ = √60

HJ = 2√15 units

And

Cos(G) = 14/16 = 7/8

∠G = cos⁻¹(7/8)

∠G = 67.5 Degrees

And

Sin(J) = 14/16 = 7/8

∠J = Sin⁻¹(7/8)

∠J = 61.04 degree

Thus, required values are HJ = 2√15 units, ∠G = 67.5 Degrees, ∠J = 61.04 degree.

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The given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

An exponential function is a mathematical function of the form f(x) = a^x, where a is a positive constant and x is any real number. The base a is typically a number greater than 1, and the function grows or decays rapidly depending on whether a is greater than or less than 1.

Exponential functions are commonly used to model processes that exhibit exponential growth or decay, such as population growth, radioactive decay, and compound interest. They also arise in various areas of mathematics and science, including calculus, probability theory, and physics.

We are given the exponential function [tex]y=3^x.[/tex]

Let x1 be any value of x, then the corresponding y-value is [tex]y1=3^{(x_1)[/tex]

Let x2=x1+1 be the next value of x, then the corresponding y-value is [tex]y2=3^x2=3^(x1+1)=3*3^x1.[/tex]

So, we can see that y2 is 3 times y1, which means the y-values increase by a factor of 3 between any two points separated by x2−x1=1.

Therefore, the given exponential function satisfies the property of increasing by a factor of 3 between any two points separated by x2−x1=1.

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Step-by-step explanation:

A, 7^(x+3) = 823, 543

7^(x+3) = 823, 543

7^(x+3) = 7^7 ....... write in the power form which it's base must be 7 in order to equalify the power

x+3 = 7

x = 4

B, 4^ -4x = 4^-8

4 ^ -4x = 1/65, 536

4^ -4x = 1/4^8 ........ write in the power form

4^ -4x = 4^-8

-4x = -8 ..... write equality among the power cause it's base is same

x=2

C, 1/(6^(x-5) ) = 1296

1/(6^(x-5) ) = 1296

6^-(x-5) = 1296

6^-(x-5) = 6^4........ write in the power form

-(x-5) = 4

-x + 5 = 4

-x = -11

x = 1

D, 1/3^x+7 = 1/243

1/3^x+7 = 1/243

3 ^ -(x+7) = 3^-5 .... write in the power form

-(x+7) = -5

-x-7 = -5

-x = 2

x= -2

The minimum number of people that need to be persuaded is two. When there are 0 contributor, 1 contributor, 2 or more contributor this is a the Nash equilibria.

a. Let's first calculate the benefit to each citizen when there are n contributors. According to the problem, the benefit is n^2 dollars. So when there are 0 contributors, the benefit to each citizen is 0 dollars. When there is 1 contributor, the benefit to each citizen is 1 dollar. When there are 2 contributors, the benefit to each citizen is 4 dollars. And so on, up to 10,000 dollars per citizen when all 100 citizens contribute.

Now let's think about the incentives of each citizen to contribute. If no one contributes, everyone gets 0 dollars of benefit. If one person contributes, that person gets 1 dollar of benefit, and everyone else gets 0 dollars. So each person has an incentive to free-ride, hoping that someone else will contribute.

But if two people contribute, each person gets 4 dollars of benefit, which is more than the 1 dollar cost of contributing. So once there are at least two contributors, it becomes rational for everyone else to contribute as well.

Therefore, the minimum number of people that need to be persuaded is two. Once two people contribute, it becomes rational for everyone else to contribute as well.

b. Let's consider the Nash equilibria of the game where each citizen is deciding whether to contribute. A Nash equilibrium is a situation where no one has an incentive to change their strategy, given the strategies of all the other players.

In this case, each citizen has two strategies: contribute or free-ride. Let's consider the case where n citizens are contributing. If everyone else is contributing, then it is rational to contribute as well, since the benefit of contributing is greater than the cost.

If everyone else is free-riding, then it is rational to free-ride as well, since the cost of contributing is greater than the benefit. However, if some people are contributing and some people are free-riding, then it may be rational to contribute, since the benefit of contributing may outweigh the cost, depending on the number of contributors.

Therefore, there are multiple Nash equilibria in this game, depending on the number of contributors. When there are 0 contributors, everyone is free-riding and this is a Nash equilibrium. When there is 1 contributor, that person is contributing and everyone else is free-riding, and this is a Nash equilibrium. When there are 2 or more contributors, everyone is contributing, and this is a Nash equilibrium.

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According to the information, and example of a counterexample is: A line can cross the circle only at one point.

To find a counterexample to this statement we must read it carefully and identify the main idea of it. In this case you are stating that a line always crosses a circle at two points.

Later we must analyze this statement and evaluate if it is true or false. In this case it is false because we can make a line that crosses a circle only once. So, the counter example sentence would be.

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(a) To minimize the area of the right triangle formed by the cable, ground, and building, we need to minimize the length of the cable. To do this, we can use the Pythagorean theorem:

c^2 = a^2 + b^2

where c is the length of the cable, a is the distance from the guidepost to the point where the cable is attached to the building, and b is the distance from that point to the ground.

Since we want to minimize c, we can differentiate the equation with respect to a and set the derivative equal to zero:

dc/da = 2a/c = 0

Solving for a, we get a = c/2. This means that the point where the cable is attached to the building should be halfway up the building, or 10 feet high.

(b) To minimize the length of the cable, we can use the principle of least action, which states that the path taken by the cable is the one that minimizes the integral of the tension along the cable.

Assuming that the tension in the cable is constant, we can use the law of sines to find the angle θ:

sin θ / 20 = sin (90° - θ) / c

where c is the length of the cable.

We want to minimize c, so we can differentiate the equation with respect to θ and set the derivative equal to zero:

d(c)/d(θ) = -20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * dc/d(θ) = 0

Solving for dc/d(θ), we get:

dc/d(θ) = 20c * tan(θ)

Substituting this into the original equation, we get:

-20cos(θ) / sin^2(θ) + cos(θ) / sin(θ) * 20c * tan(θ) = 0

Simplifying, we get:

cos(θ) / sin(θ) = tan(θ)

Solving for θ, we get:

θ = 45°

Therefore, to minimize the length of the cable, it should make an angle of 45° with the face of the building.

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Answer:

Based on the given information, we can conclude that the graph represents a quadratic function. The vertex of the parabola is located at (0, 9) and the function passes through several points including (-4, -7), (-3, 0), (3, 0), and (4, -7).

To find the equation of the function, we need to determine the value of "a" in the equation f(x) = ax^2 + bx + c. Since the vertex is located at (0, 9), we know that the x-coordinate of the vertex is 0. Therefore, we can use the vertex form of the equation, which is f(x) = a(x - 0)^2 + 9, or simply f(x) = ax^2 + 9.

Next, we can use one of the given points to solve for "a". Let's use the point (-3, 0).

0 = a(-3)^2 + 9

0 = 9a - 9

9 = 9a

a = 1

Therefore, the value of "a" in the equation of the function is B. 1.

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The event that has a probability of 0 is selecting a yellow marble

Other probabilities are listed below

The items in the jar are given as

6 black, 4 brown, 8 white, and 2 blue.

These items are the sample space of this event

Hence, the sample space is 6 black, 4 brown, 8 white, and 2 blue.

For the probability Julie will draw a white marble, we have

P(White) = White/Total

So, we have

P(White) = 8/20

Simplify

P(White) = 2/5

For the event that is more likely to happen, we have

P(black marble) = 6/20

P(brown or blue marble) = (4 + 2)/20

P(brown or blue marble) = 6/20

The probabilities are equal

So, both events have equal likelihood

The event that has a probability of 0 could be the probability of selecting a yellow marble

This is because the jar has no yellow marble

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(a) Identifying the population, parameter, sample, and statistic for a study on the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts before and after a campaign.

(b) Stating the null and alternative hypotheses for a significance test on whether the percentage of male drivers not using seatbelts has decreased after the campaign.

(a) Population: All male drivers between the ages of 19 and 29 in the major urban area.

Parameter: The percentage of male drivers between the ages of 19 and 29 in the major urban area who do not regularly use seatbelts after the radio and television campaign and stricter enforcement by the local police.

Sample: 100 male drivers between the ages of 19 and 29 who were polled.

Statistic: The percentage of male drivers between the ages of 19 and 29 in the sample who did not wear their seatbelts, which is 24%.

(b) The null hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has not decreased after the radio and television campaign and stricter enforcement by the local police.

The alternative hypothesis is that the percentage of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area has decreased after the radio and television campaign and stricter enforcement by the local police.

Mathematically, the hypotheses can be stated as follows:

H0: p >= 0.28

Ha: p < 0.28

where p is the proportion of male drivers between the ages of 19 and 29 who do not regularly use seatbelts in the major urban area after the radio and television campaign and stricter enforcement by the local police.

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The question is -

In a major urban area, the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts was 28%. After a major radio and television campaign and stricter enforcement by the local police, researchers want to know if the percentage of male drivers between the ages of 19 and 29 who did not regularly use seatbelts has decreased.  They polled a random sample of 100 males between the ages of 19 and 29 and find the percentage who didn’t wear their seatbelts was 24%.

(a) Identify the population, parameter, sample, and statistic.

(b) State appropriate hypotheses for performing a significance test.

If Sarah uses 3/4 yard of ribbon to make a hair bow, she will need 6 and 3/4 yards of ribbon to make 9 hair bows.

To find out how many yards of ribbon Sarah will use to make 9 hair bows, we need to multiply the amount of ribbon used for one hair bow (3/4 yard) by the number of hair bows she wants to make (9).

So, the equation we need to use is:

3/4 yard of ribbon per hair bow x 9 hair bows = ? yards of ribbon

To solve for the answer, we can simplify the equation:

3/4 x 9 = 27/4

So Sarah will need 27/4 yards of ribbon to make 9 hair bows.

To convert this fraction to a mixed number, we can divide the numerator (27) by the denominator (4) and write the remainder as a fraction:

27 ÷ 4 = 6 with a remainder of 3

In summary, Sarah will need 6 and 3/4 yards of ribbon to make 9 hair bows, if she uses 3/4 yard of ribbon to make one hair bow.

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

To prove √(6a+3) <a + 2 for all a > 1 using the Mean Value Theorem, we will begin by defining a function f(x) as:

f(x) = √(6x+3)

We can see that f(x) is a continuous and differentiable function for all x > -1/2.

Now, let's choose two values of a, such that a > 1 and b = a + h, where h is a positive number. By the Mean Value Theorem, there exists a value c between a and b such that

f(b) - f(a) = f'(c)(b-a)

where f'(c) is the derivative of f(x) evaluated at c.

Now, let's evaluate the derivative of f(x) as:

f'(x) = 3/(√(6x+3))

Thus, we can write

f(b) - f(a) = f'(c)(b-a)

√(6(a+h)+3) - √(6a+3) = f'(c)h

Dividing both sides by h and taking the limit as h → 0, we get

lim h→0 (√(6(a+h)+3) - √(6a+3))/h = f'(a)

Now, we can evaluate the limit on the left-hand side using L'Hopital's rule

lim h→0 (√(6(a+h)+3) - √(6a+3))/h = lim h→0 [3/(√(6(a+h)+3)) - 3/(√(6a+3))] = 3/(2√(6a+3))

Therefore, we have

f'(a) = 3/(2√(6a+3))

Now, we can use this value to rewrite the inequality as

√(6a+3) - (a + 2) < 0

Multiplying both sides by 2√(6a+3) and simplifying, we get

3 < 4a + 2√(6a+3)

Subtracting 4a from both sides and squaring, we get

9 < 16a^2 + 16a + 24a + 12

Simplifying, we get

0 < 16a^2 + 40a + 3

This inequality holds for all a > 1, so we have proved that

√(6a+3) < a + 2 for all a > 1.

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The given question is incomplete, the complete question is:

Use Mean value theorem to prove √(6a+3) <a + 2 for all a > 1. Using methods other than the Mean Value Theorem will yield

The direction angles of the vector u = (-1, 9, -6) are approximately -6.1°, 64.8°, and -75.2° for the x, y, and z axes, respectively.

The direction angles of a vector are the angles that the vector makes with the positive x, y, and z axes. To find the direction angles of the vector u = (-1, 9, -6), we can use the formulas:

cosθx = u_x/||u||, cosθy = u_y/||u||, and cosθz = u_z/||u||

where θx, θy, and θz are the angles that u makes with the x, y, and z axes, respectively, and ||u|| is the magnitude of u, given by:

||u|| = √(u_x² + u_y² + u_z²)

Substituting the values of u, we have:

||u|| = √((-1)² + 9² + (-6)²) = √118

cosθx = -1/√118 ≈ -0.183, cosθy = 9/√118 ≈ 0.551, and cosθz = -6/√118 ≈ -0.366

Taking the inverse cosine of each of these values, we get:

θx ≈ -6.1°, θy ≈ 64.8°, and θz ≈ -75.2°

Therefore, the direction angles of the vector u are approximately -6.1°, 64.8°, and -75.2° for the x, y, and z axes, respectively.

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The probability of hitting the bulls eye (center circle) when we shoot an arrow with radius of the bulls eye is 2ft and the radius of the target is 6ft is 11.1%.

Assuming that we are shooting randomly at the target and the arrow can land anywhere within the target area, the probability of hitting the bull's eye (center circle) can be calculated as the ratio of the area of the bull's eye to the area of the entire target.

The area of the bull's eye can be calculated as follows:

Area of bull's eye = π x (radius of bull's eye)²
Area of bull's eye = 3.14 x 2²
Area of bull's eye = 12.56 square feet

The area of the entire target can be calculated as follows:

Area of target = π x (radius of target)²
Area of target = 3.14 x 6²
Area of target = 113.04 square feet

Therefore, the probability of hitting the bull's eye can be calculated as:

Probability of hitting bull's eye = Area of bull's eye / Area of target
Probability of hitting bulls eye = 12.56 / 113.04
Probability of hitting bulls eye = 0.111 or approximately 11.1%

So, the probability of hitting the bull's eye (center circle) when we shoot an arrow randomly at the target is approximately 11.1%.

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Answer:

Using the Distance Formula:

EG =√((-a - b)^2) + (0 - c)^2)

=√((a + b)^2 + c^2)

FH =√((-b - a)^2 + (c - 0)^2)

=√((a + b)^2 + c^2)

So EG = FH.

The amount in pounds of kale Lori bought is 1 1/2 pounds.

To find out how many pounds of kale Lori bought, you need to multiply the total weight of the vegetables by the fraction that represents the proportion of kale.

In this case, you can calculate the amount of kale as follows:

(7 1/2) * (1/5)

First, convert the mixed number 7 1/2 to an improper fraction:

(7 * 2 + 1) / 2 = 15/2

Now multiply the two fractions:

(15/2) * (1/5)

Multiply the numerators together and the denominators together:

(15 * 1) / (2 * 5) = 15 / 10

Now, simplify the fraction:

15 ÷ 5 / 10 ÷ 5 = 3 / 2

So, Lori bought 3/2 or 1 1/2 pounds of kale from the grocery store. This means that out of the total 7 1/2 pounds of vegetables she purchased, 1 1/2 pounds were kale.

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The measure of side OP in quadrilateral OPQR is 0.

To find the measure of side OP in quadrilateral OPQR, which is similar to quadrilateral KLMN, follow these steps:

1. Identify the corresponding sides in both quadrilaterals. In this case, side OP corresponds to side KL, side OQ corresponds to side KM, side PQ corresponds to side LN, and side QR corresponds to side MN.

2. Determine the scale factor between the quadrilaterals by comparing the lengths of corresponding sides. Since we have the lengths of sides KM (13), LN (27), and MN (57), we can use the ratio of KM/LN (13/27) or MN/LN (57/27) as the scale factor.

3. Apply the scale factor to the length of side KL (0) to find the length of side OP. Since the length of side KL is 0, multiplying by the scale factor (either 13/27 or 57/27) will still result in a length of 0 for side OP.

4. Round your answer to the nearest tenth if necessary. In this case, the length of side OP is 0, so rounding is not necessary.

So, the measure of side OP in quadrilateral OPQR is 0.

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Answer:

The data in the table represent an exponential function.

[tex]y = 2( {3}^{x} )[/tex]

The number of left-handed students in the first trial of the simulation can be found by following the above steps and counting the occurrences of two-digit numbers within the 00-14 range on line 1 of the random-number table.

To find out how many left-handed students occur in the first trial of the simulation, you'll need to follow these steps,
1. Identify the probability range for left-handed students, which is 00-14 as you've mentioned.
2. Start on line 1 of the random-number table.
3. Read each two-digit number on the line and check if it falls within the range 00-14.
4. Count the number of times a number within the 00-14 range appears in the first 10 two-digit numbers (since you're selecting a random sample of 10 students).
5. The count of numbers within the 00-14 range represents the number of left-handed students in the first trial of the simulation.

By doing the aforementioned processes and counting the occurrences of two-digit numbers between the ranges of 00 and 14 on line 1 of the random-number table, it is possible to determine the number of left-handed pupils in the simulation's first trial.

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The maximum height of the arrow is 105 feet. To find the maximum height of the arrow, we need to determine the vertex of the quadratic function h = -16[tex]t^{2}[/tex] + 80t + 25.

The vertex is the highest point on the graph of the function, which represents the maximum height of the arrow.

To find the t-value at the vertex, we use the formula t = -b/2a, where a = -16 and b = 80. Plugging these values into the formula gives us t = -80/(2(-16)) = 2.5 seconds.

To find the maximum height, we plug t = 2.5 into the equation to get h = -16[tex](2.5)^{2}[/tex] + 80(2.5) + 25 = 105 feet. Therefore, the maximum height of the arrow is 105 feet.

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Answer:

Step-by-step explanation:

AngleSideSide because it's bad

but also, if you had an angle, a side and a side

For your example: let's say CD≅AS

You could change the angle of S or D and the parameters of the triangle would still be true.   Because you can change something and still have AngleSideSide be true, would make them not congruent any more.

Answer:

3

Step-by-step explanation:

We can see that in figure ABC, line segment AB is 5 ft.

We can also see that in figure DEF, line segment DE is 15 ft.

How did we get from 5 to 15?

We multiplied by 3, so the scale factor is 3.

Hope this helps! :)

The critical numbers of f(x) are x = -1, 0, and 1 and The sum of all critical numbers is 0.

To find the critical numbers of f(x) = x²/4(x+1), we need to find the values of x where the derivative of f(x) is equal to zero or does not exist.

The derivative of f(x) is:

f'(x) = [(x+1)(2x) - x²(4)] / [4(x+1)²]

Simplifying, we get:

f'(x) = [2x(x+1) - 4x²] / [4(x+1)²]

f'(x) = [2x(x+1-2x)] / [4(x+1)²]

f'(x) = [2x(1-x)] / [4(x+1)²]

f'(x) = [x(1-x)] / [2(x+1)²]

The critical numbers are the values of x where f'(x) is equal to zero or does not exist.

Setting f'(x) = 0, we get:

x(1-x) = 0

This equation is true when x = 0 or x = 1.

Now, let's check if f'(x) does not exist at x = -1 (which is the only possible point where the derivative may not exist):

f'(x) = [x(1-x)] / [2(x+1)²]

When x = -1, the denominator of f'(x) becomes zero, so the derivative does not exist at x = -1.

Therefore, the critical numbers of f(x) are x = -1, 0, and 1.

The sum of all critical numbers is:

-1 + 0 + 1 = 0

Hence, the answer is 0.

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To calculate the cost of each tile, we need to first determine the volume of each tile. The length and width of the tile are given as 16 inches, and the thickness is given as 1/4 inches, which can be converted to 0.25 inches. Therefore, the volume of each tile is 16 x 16 x 0.25 = 64 cubic inches.

Next, we need to determine the weight of gold in each tile. Since the density of gold is 11.17 ounces per cubic inch, the weight of gold in each tile is 64 x 11.17 = 715.68 ounces.

Finally, we can calculate the cost of each tile by multiplying the weight of gold by the price of gold per ounce. The price of gold is given as $1,303.80 per ounce, so the cost of each tile is 715.68 x $1,303.80 = $933,526.78. Rounded to the nearest dollar, each tile will cost $933,527.

In summary, each square tile made of solid gold and measuring 16 inches long and 1/4 inches thick will cost approximately $933,527. This cost is based on the density of gold, which is 11.17 ounces per cubic inch, and the price of gold, which is $1,303.80 per ounce.

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Let the fraction be x/y.

We know that x + y = 12, and that (x) / (y + 3) = 12.

Multiplying both sides of the second equation by (y + 3), we get:

x = 12(y + 3)

Substituting this into the first equation, we get:

12(y + 3) + y = 12

Expanding and simplifying, we get:

13y + 36 = 12

Subtracting 36 from both sides, we get:

13y = -24

Dividing both sides by 13, we get:

y = -24/13

Substituting this value of y into the equation x + y = 12, we get:

x - 24/13 = 12

Multiplying both sides by 13, we get:

13x - 24 = 156

Adding 24 to both sides, we get:

13x = 180

Dividing both sides by 13, we get:

x = 180/13

Therefore, the fraction is 180/13 divided by -24/13, which simplifies to -15/2.

To evaluate the limit using L'Hospital's rule, we need to take the derivative of both the numerator and denominator separately until we get a determinate form. We have:

lim (e^x + 2x - 1) / (2x)

Taking the derivative of the numerator:

lim (e^x + 2) / 2

Taking the derivative of the denominator:

lim 2

Since we now have a determinate form, we can evaluate the limit by plugging in the value of x. We get:

(e^x + 2) / 2

As x approaches infinity, e^x also approaches infinity, so the limit diverges to positive infinity. Therefore, the limit does not exist.

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Given that quadrilateral PQRS is a parallelogram, you can prove that it is also a rectangle by A: Use the distance formula to find the length of both diagonals to see if they are congruent and B: Find the slopes of all sides to determine if the angles are right angles. Therefore, the correct option is C: C. Both A and B are valid.

To prove that PQRS is a rectangle, we need to show that all angles are right angles.

Option A: Using the distance formula, we can find the lengths of both diagonals, PR and QS. If PR and QS are congruent, then we know that opposite sides of the parallelogram are congruent and parallel (since PQRS is a parallelogram). If we can also show that PR and QS intersect at a 90-degree angle, then we have proven that PQRS is a rectangle.

Option B: Finding the slopes of all sides can help us determine if the angles are right angles. If the product of the slopes of adjacent sides is -1, then we know that the sides are perpendicular (since the slope of a line perpendicular to another line is the negative reciprocal of its slope). If we can show that all adjacent sides have slopes that multiply to -1, then we have proven that PQRS is a rectangle.

Both options A and B can be used to prove that PQRS is a rectangle, so the correct answer is C.

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The estimated areas of each curve are listed below:

Case 1: A = 12.75

Case 2: A = 12.5

In this problem we must estimate the area above the x-axis and under a curve by using sums of rectangles and triangles according to the following expression:

A = {∑ [MIN (f(xₙ₋₁), f(xₙ))] + 0.5 · ∑ [MAX (f(xₙ₋₁), f(xₙ)) - MIN (f(xₙ₋₁), f(xₙ))]} · Δx, for n = {1, 2, 3, ..., N}

Where:

Case 1

A = (3.5 + 3.5 + 1.5) · 1 + 0.5 · (3.5 + 0.75 + 0.75 + 2 + 1.5) · 1

A = 8.5 + 0.5 · 8.5

A = 12.75

Case 2

A = (1 + 3 + 4) · 1 + 0.5 · (1 + 2 + 1.5 + 0.5 + 4) · 1

A = 8 + 4.5

A = 12.5

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