Question

look at the image. given the dimensions in the drawing, what is the approximate area of the figure?

Explanation:

Step1: Calculate the area of the rectangle

The formula for the area of a rectangle is \(A = l\times w\). Here, \(l = 96\) in and \(w = 32\) in. So, \(A_{rectangle}=96\times32 = 3072\) \(in^{2}\).

Step2: Calculate the area of the right - angled triangle

First, find the base of the right - angled triangle. Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(c = 30\) in and \(a = 24\) in. Let the base \(b=\sqrt{30^{2}-24^{2}}=\sqrt{(30 + 24)(30 - 24)}=\sqrt{54\times6}=\sqrt{324}=18\) in.
The formula for the area of a triangle is \(A=\frac{1}{2}\times base\times height\). Here, base \(b = 18\) in and height \(h = 24\) in. So, \(A_{triangle}=\frac{1}{2}\times18\times24= 216\) \(in^{2}\).

Step3: Calculate the area of the semicircle

The diameter of the semicircle \(d = 30\) in, so the radius \(r=\frac{d}{2}=15\) in. The formula for the area of a semicircle is \(A=\frac{1}{2}\pi r^{2}\). Using \(\pi\approx3.14\), \(A_{semicircle}=\frac{1}{2}\times3.14\times15^{2}=\frac{1}{2}\times3.14\times225 = 353.25\) \(in^{2}\).

Step4: Calculate the total area

The total area \(A = A_{rectangle}+A_{triangle}+A_{semicircle}\).
\(A=3072 + 216+353.25=3641.25\approx3641\) \(in^{2}\).

Answer:

The approximate area of the figure is \(3641\) \(in^{2}\).