Question

the perimeter of a rectangle is 36 cm. the rectangles area (in sq. cm) as a function of its length (in cm) is graphed. what is the approximate average rate at which the area decreases, as the rectangles length goes from 13 cm to 16 cm? choose 1 answer: 9 1/3 sq. cm per cm 9 2/3 sq. cm per cm 10 1/3 sq. cm per cm 11 1/3 sq. cm per cm

Explanation:

Step1: Recall the formula for average rate of change

The formula for the average rate of change of a function \(y = f(x)\) from \(x = a\) to \(x = b\) is \(\frac{f(b)-f(a)}{b - a}\). Here, \(x\) is the length of the rectangle and \(y\) is the area of the rectangle. So, \(a = 13\), \(b=16\).

Step2: Find the area values from the graph

From the graph, when \(x = 13\) (length \(l = 13\) cm), the area \(A(13)\approx64\) sq.cm. When \(x = 16\) (length \(l = 16\) cm), the area \(A(16)\approx36\) sq.cm.

Step3: Calculate the average rate of change

Substitute into the formula \(\frac{A(16)-A(13)}{16 - 13}\). We have \(\frac{36 - 64}{16-13}=\frac{- 28}{3}=-9\frac{1}{3}\). The magnitude (since we are asked about the rate at which the area decreases) is \(9\frac{1}{3}\) sq.cm per cm.

Answer:

A. \(9\frac{1}{3}\) sq. cm per cm