Question

the area of a certain rectangle varies directly to the square of its shorter side. if the length of the shorter side is 4 cm, then the area is 160 cm².
a) write an equation that relates the area of the rectangle to the squared length of the shortest side. you may define your own single - letter variables.
b) what is the area of the rectangle if the length of the shortest side is 7?

Explanation:

Step1: Define variables and direct - variation formula

Let \(A\) be the area of the rectangle and \(x\) be the length of the shorter side. Since \(A\) varies directly as \(x^{2}\), the formula is \(A = kx^{2}\), where \(k\) is the constant of variation.

Step2: Find the value of \(k\)

Given \(x = 4\) and \(A=160\). Substitute into \(A = kx^{2}\):
\(160=k\times4^{2}\)
\(160 = 16k\)
\(k=\frac{160}{16}=10\)

Step3: Write the equation

The equation is \(A = 10x^{2}\)

Step4: Find the area when \(x = 7\)

Substitute \(x = 7\) into \(A = 10x^{2}\)
\(A=10\times7^{2}\)
\(A = 10\times49\)
\(A=490\)

Answer:

a) The equation is \(A = 10x^{2}\)
b) The area is \(490\space cm^{2}\)