Question

A contractor is constructing the house shown in the figure. The cross section
up to the roof is in the shape of a rectangle. The area of the rectangular floor
of the house is 1500 square feet. Express the perimeter of the rectangular
floor, P, as a function of the width of the rectangle, X.

Answer 1

The perimeter of the rectangular floor is the sum of its dimensions

The perimeter function is $\mathbf{P(x) = \frac{2x^2 + 3000}x}$

The area of the rectangular floor is given as: 1500

The area of a rectangle is calculated as:

$\mathbf{Area = xy}$

Where: x and y represents the width and the length of the floor, respectively.

So, we have:

$\mathbf{xy = 1500}$

Make y the subject

$\mathbf{y = \frac{1500}x}$

The perimeter of the rectangular floor is:

$\mathbf{P = 2(x + y)}$

Substitute $\mathbf{y = \frac{1500}x}$

$\mathbf{P = 2(x + \frac{1500}x)}$

Take LCM

$\mathbf{P = 2(\frac{x^2 + 1500}x)}$

Open bracket

$\mathbf{P = \frac{2x^2 + 3000}x}$

Express as a function

$\mathbf{P(x) = \frac{2x^2 + 3000}x}$

Hence, the perimeter function is $\mathbf{P(x) = \frac{2x^2 + 3000}x}$

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