Question

you are given a rectangle with perimeter p = 36 and sides x and y. as side x increases, how will the area of the rectangle change? a the area of the rectangle will decrease, because the perimeter stays the same and therefore y decreases. b the area of the rectangle will increase until x becomes 9. as x increases further, the area will decrease. c the area of the rectangle will increase, because x increases.

Explanation:

1. First, recall the perimeter formula for a rectangle: \( P = 2(x + y) \). Given \( P = 36 \), we have \( 2(x + y)=36 \), so \( x + y = 18 \), and \( y=18 - x \).
2. The area of a rectangle is \( A=xy=x(18 - x)=-x^{2}+18x \). This is a quadratic function with \( a=- 1<0 \), so its graph is a parabola opening downward. The vertex of the parabola \( y = ax^{2}+bx + c \) is at \( x=-\frac{b}{2a} \). For \( A=-x^{2}+18x \), \( a=-1 \), \( b = 18 \), so \( x=-\frac{18}{2\times(-1)} = 9 \). At \( x = 9 \), the area is maximized. So as \( x \) increases from 0 to 9, the area increases, and as \( x \) increases beyond 9, the area decreases.

Answer:

B. The area of the rectangle will increase until \( x \) becomes 9. As \( x \) increases further, the area will decrease.