Question

if ( a = 4 ) in and ( b = 3 ) in, what is the area of the sketch?
note: figure not drawn to scale
a. ( 60 mathrm{in}^{2} )
b. ( 84 mathrm{in}^{2} )
c. ( 144 mathrm{in}^{2} )
d. ( 39.5 mathrm{in}^{2} )

Explanation:

Step1: Calculate the area of the rectangle

The formula for the area of a rectangle is \(A_{rectangle} = length\times width\). Here, the length is \(4b\) and the width is \(b\). Substitute \(b = 3\) into the formula: \(A_{rectangle}=4b\times b=4\times3\times3 = 36\) \(in^{2}\).

Step2: Calculate the area of the trapezoid

The formula for the area of a trapezoid is \(A_{trapezoid}=\frac{(a_{1} + a_{2})h}{2}\). Here, \(a_{1}=a\), \(a_{2}=2a\), and \(h = a\). Substitute \(a = 4\) into the formula: \(A_{trapezoid}=\frac{(a + 2a)\times a}{2}=\frac{3a\times a}{2}=\frac{3\times4\times4}{2}=24\) \(in^{2}\).

Step3: Calculate the total area

The total area \(A = A_{rectangle}+A_{trapezoid}\). Substitute the values of \(A_{rectangle}\) and \(A_{trapezoid}\): \(A=36 + 24=60\) \(in^{2}\).

Answer:

A. \(60\ in^{2}\)