Question

There are two buildings that you want to have in the amusement park, but the size hasn’t been determined yet. Although you don’t know the specific dimensions, you do know the relationships between the sides. The first is the rectangular gift shop. You know that the length will be 20x+24 feet and the width will be 36x-20 feet. a) Write the expression that represents the area of the gift shop, in terms of x. b) Write the expression that represents the perimeter of the gift shop, in terms of x. c) If the perimeter is going to be 176 feet, what are the dimensions of the building?

Answer 1

A) The area of a rectangle is A = lw, where l=length of the rectangle and w=width of the rectangle. You know the length of the gift shop, l = 20x + 24. You know the width, w = 36x - 20. Plug those expressions into the equation for area of a rectangle and multiply/foil:
$A = lw\\ A = (20x + 24)(36x - 20)\\ A = (20x)(36x) + (20x)(-20) + (24)(36x) + (24)(-20)\\ A = 720x^{2} - 400x + 864x - 480\\ A = 720x^{2} + 464x - 480$

The expression for the area of the gift shop is $720x^{2} + 464x - 480$.

B) The equation for the perimeter of the gift shop is P = 2(l+w), where l = length and w = width. Plug your values for l and w into this equation:
$P = 2(l+w)\\ P = 2(20x + 24 + 36x - 20)\\ P = 2(56x + 4) = 2(56x) + 2(4)\\ P = 112x + 8$

The expression for the perimeter of the gift shop is 112x + 8

C) Since you know the perimeter is going to be 176 ft, that means P = 176. Plug that into the equation you found in part B, P = 112x + 8, and solve for x.
$P = 112x + 8\\ 176 = 112x + 8\\ 112x = 168 x = 1.5$

Once you solve for x, you can plug x into your equations for width and length to find the dimensions. x = 1.5, so:
1) Length = 20x+24 feet 
Length = 20(1.5) + 24 feet = 54 feet
2) Width = 36x-20 feet
Width = 36(1.5)-20 feet = 34 feet

Your dimensions are 54 feet (length) by 34 feet (width).