Question

select the correct answer
janis is the set designer for a play. she has 40 feet of dividers to use to create a section in the workshop for storing props. she plans to create the space in the corner of the workshop and use the dividers to create the extra two walls. the area of the sectioned off space, a, is modeled by this equation, where x is the width of the prop section.
$a = -x^2 + 40x$
which equation reveals the dimensions that will create the maximum area of the prop section?
a. $a = -x(x - 40)$
b. $a = -(x - 20)^2 + 400$
c. $a = -(x - 20)(x - 20)$
d. $a = -(x - 400)^2$

Explanation:

Step1: Recall vertex form of quadratic

A quadratic \( y = ax^2 + bx + c \) in vertex form is \( y = a(x - h)^2 + k \), where \((h,k)\) is the vertex. For maximum area, we need vertex form (since \( a=-1<0 \), parabola opens down, vertex is maximum).

Step2: Complete the square for \( A = -x^2 + 40x \)

Factor out \(-1\) from the first two terms: \( A = - (x^2 - 40x) \).
To complete the square inside the parentheses, take half of \(-40\) (which is \(-20\)), square it (\( (-20)^2 = 400 \)), and add/subtract:
\( A = - (x^2 - 40x + 400 - 400) \)
\( A = - [(x - 20)^2 - 400] \)
Distribute the \(-1\): \( A = - (x - 20)^2 + 400 \).

Answer:

B. \( A = -(x - 20)^2 + 400 \)