Q: Given the diagram, if the area $$A is $$given by $$A = x y$$ and the total fencing used is $$y + 2x = 2000$$, what is the maximum possible area for the picnic enclosure?

The maximum area is $$A = 500 \times 1000 = 500{,}000$$ square yards.

Question 1

A rectangular picnic area is to be built along a road, with no fence required along the road itself. The area is enclosed by fencing on the other three sides, as shown in the diagram. If 2000 yards of fencing material are available, what are the dimensions that maximize the area of the picnic enclosure?

Accepted Answer

The width along the road should be $$y = 1000$$ yards and the depth perpendicular to the road should be $$x = 500$$ yards.

Question 2

Given the diagram, if the area $$A is $$given by $$A = x y$$ and the total fencing used is $$y + 2x = 2000$$, what is the maximum possible area for the picnic enclosure?

Accepted Answer

The maximum area is $$A = 500 	imes 1000 = 500{,}000$$ square yards.

Question 3

If $$x is $$the length of the sides perpendicular to the road and $$y is $$the length along the road, which equation correctly expresses the constraint for the fencing material?

Accepted Answer

$$y + 2x = 2000$$

Question 4

To maximize the area $$A = x y$$ given the constraint $$y + 2x = 2000$$, what is the value of $$x at $$the maximum area?

Accepted Answer

$$x = 500$$

Question 5

What calculus method is used to solve for the maximum area of the picnic enclosure given the constraint $$y + 2x = 2000$$?

Accepted Answer

Substitute $$y = 2000 - 2x$$ into $$A = x y$$, then take the derivative of $$A$$ with respect to $$x$$, set it to zero, and solve for $$x.$$

Question 6

If the area is $$A = x y$$ and the constraint is $$y + 2x = 2000$$, what is the expression for $$A in $$terms of $$x$$ only?

Accepted Answer

$$A = x (2000 - 2x)$$

Question 7

After substituting $$y = 2000 - 2x$$ into $$A = x y$$, what is the derivative $$\frac{dA}{dx}$$?

Accepted Answer

$$\frac{dA}{dx} = 2000 - 4x$$

Question 8

What is the value of $$x$$ that maximizes the area, based on setting $$\frac{dA}{dx} = 0$$?

Accepted Answer

$$x = 500$$

Question 9

If the maximum area is $$500{,}000$$ square yards, what is the length of fencing used along the road?

Accepted Answer

$$y = 1000$$ yards

Question 10

If the area is maximized when $$x = 500$$ yards and $$y = 1000$$ yards, what is the ratio of the length along the road to the depth perpendicular to the road?

Accepted Answer

The ratio is $$\frac{y}{x} = 2$$

Maximizing Area of a Rectangular Enclosure (Optimization Problem)

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