**Problem Statement:**A 1000-square-foot rectangular plot of land is going to be divided into three equal-sized, adjacent playgrounds (see diagram). Find the number of feet of fencing required as a function of $ x $. (Note: there is only one fence between the playgrounds, not two.)**Diagram Explanation:**In the provided diagram, a large rectangle is shown, representing the 1000-square-foot plot of land. This rectangle is divided into three thinner, adjacent rectangles. The height of the plot is labeled as $ y $, and the combined width of the three playgrounds (entire length of the large rectangle) is labeled as $ x $.**Calculation:**To find the number of feet of fencing required as a function of $ x $, we need to determine the dimensions of the rectangle and consider the fencing needed for the entire boundary and the internal division fencing. Since it is mentioned that the area of the plot is 1000 square feet,\[ x \times y = 1000 \]\[ y = \frac{1000}{x} \]Next, we need the fencing for both the perimeter and the internal division:Perimeter fencing includes:\[ 2 \times y \] (for the top and bottom sides)\[ 2 \times x \] (for the left and right sides)\[ \text{Perimeter fencing} = 2x + 2y \]Internal division fencing includes:\[ 2 \times y \] (since there is just one fence going down through the height)Thus, the total fencing required (including both the perimeter and the internal division) is:\[ F = (2x + 2y) + 2y \]\[ F = 2x + 4y \]Substitute $ y $ from the area equation:\[ y = \frac{1000}{x} \]\[ F = 2x + 4 \left(\frac{1000}{x}\right) \]\[ F = 2x + \frac{4000}{x} \]**Answer:**\[ F(x) = 2x + \frac{4000}{x} \]This function $ F(x) $ represents the number of feet of fencing required as a function of $ x $.

The area of the plot is given. Using this you can find y in terms of x. Now you need to find the…

A 1000-square-foot rectangular plot of land is going to be divided into three equal-sized, adjacent playgrounds (see diagram). Find the number of feet of fencing required as a function of x. (Note: there is only one fence between the playgrounds, not two.) Answer: ||

A 1000-square-foot rectangular plot of land is going to be divided into three equal-sized, adjacent playgrounds (see diagram). Find the number of feet of fencing required as a function of x. (Note: there is only one fence between the playgrounds, not two.) Answer: ||

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

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**Problem Statement:**

A 1000-square-foot rectangular plot of land is going to be divided into three equal-sized, adjacent playgrounds (see diagram). Find the number of feet of fencing required as a function of $ x $. (Note: there is only one fence between the playgrounds, not two.)

**Diagram Explanation:**

In the provided diagram, a large rectangle is shown, representing the 1000-square-foot plot of land. This rectangle is divided into three thinner, adjacent rectangles. The height of the plot is labeled as $ y $, and the combined width of the three playgrounds (entire length of the large rectangle) is labeled as $ x $.

**Calculation:**

To find the number of feet of fencing required as a function of $ x $, we need to determine the dimensions of the rectangle and consider the fencing needed for the entire boundary and the internal division fencing.

Since it is mentioned that the area of the plot is 1000 square feet,

\[ x \times y = 1000 \]
\[ y = \frac{1000}{x} \]

Next, we need the fencing for both the perimeter and the internal division:

Perimeter fencing includes:
\[ 2 \times y \] (for the top and bottom sides)
\[ 2 \times x \] (for the left and right sides)

\[ \text{Perimeter fencing} = 2x + 2y \]

Internal division fencing includes:
\[ 2 \times y \] (since there is just one fence going down through the height)

Thus, the total fencing required (including both the perimeter and the internal division) is:
\[ F = (2x + 2y) + 2y \]

\[ F = 2x + 4y \]

Substitute $ y $ from the area equation:
\[ y = \frac{1000}{x} \]

\[ F = 2x + 4 \left(\frac{1000}{x}\right) \]
\[ F = 2x + \frac{4000}{x} \]

**Answer:**

\[ F(x) = 2x + \frac{4000}{x} \]

This function $ F(x) $ represents the number of feet of fencing required as a function of $ x $.

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