Question

#6 application word problem: read problem!
you are building a rectangular brick patio. the length of the patio is ((x + 11)) feet.

a) the total area of the patio can be represented by (x^2 + 14x + 33). write an equation that can be used to determine the width of the patio.
recall: area of a rectangle = (length)(width)

b) determine the width of the patio in terms of (x).

c) write an expression for the perimeter (distance around) the patio. simplify your answer as much as possible.

d) assume the perimeter of the patio is 240 feet. determine the length and width of the patio.
the length is ______ feet
the width is ______ feet

Explanation:

Step1: Set up width equation

Let $w$ = width. Use area formula:
$$w = \frac{x^2 + 14x + 33}{x + 11}$$

Step2: Factor numerator for width

Factor quadratic numerator:
$$x^2 + 14x + 33 = (x + 11)(x + 3)$$
Substitute and simplify:
$$w = \frac{(x + 11)(x + 3)}{x + 11} = x + 3$$

Step3: Derive perimeter expression

Use perimeter formula $P=2(l+w)$:
$$P = 2[(x + 11) + (x + 3)]$$
Simplify inside parentheses first:
$$P = 2(2x + 14) = 4x + 28$$

Step4: Solve for $x$ with given perimeter

Set $4x + 28 = 240$, solve for $x$:
$$4x = 240 - 28 = 212$$
$$x = \frac{212}{4} = 53$$
Calculate length and width:
Length: $x + 11 = 53 + 11 = 64$
Width: $x + 3 = 53 + 3 = 56$

Answer:

a) $\boldsymbol{w = \frac{x^2 + 14x + 33}{x + 11}}$
b) $\boldsymbol{x + 3}$
c) $\boldsymbol{4x + 28}$
d) Length: $\boldsymbol{64}$ feet, Width: $\boldsymbol{56}$ feet