Question

problème 3: détermine l’aire de la figure sachant que la circonférence du demi-cercle mesure ( 8pi ) cm, que le rapport entre la longueur de la base et le rayon du demi-cercle est 3:1 et que la hauteur de la figure correspond à ( \frac{3}{4} ) de la mesure du diamètre du demi-cercle.

Explanation:

Step1: Find the radius of the semicircle

The formula for the circumference of a semicircle is \( C = \pi r+ 2r=( \pi + 2)r\)? Wait, no, the circumference of a semicircle (the curved part plus the diameter? Wait, no, in the context of the figure, the curved part is the arc length of the semicircle. The arc length of a semicircle is \( l=\pi r\). We are given that the circumference (arc length) of the semicircle is \( 8\pi\) cm. So we set up the equation:
\(\pi r=8\pi\)
Divide both sides by \(\pi\):
\(r = 8\) cm

Step2: Find the base of the trapezoid

The ratio of the base of the trapezoid to the radius of the semicircle is \(3:1\). Let the base be \(b\) and radius be \(r\). So \(\frac{b}{r}=\frac{3}{1}\), since \(r = 8\) cm, then \(b=3\times r=3\times8 = 24\) cm

Step3: Find the height of the trapezoid

The height of the figure (trapezoid) is \(\frac{3}{4}\) of the diameter of the semicircle. The diameter \(d = 2r=2\times8 = 16\) cm. So the height \(h=\frac{3}{4}\times d=\frac{3}{4}\times16 = 12\) cm

Step4: Find the area of the trapezoid

The formula for the area of a trapezoid is \(A_{trapezoid}=\frac{(a + b)h}{2}\), where \(a\) is the top - length (which is equal to the length of the diameter of the semicircle, \(a = d=16\) cm), \(b = 24\) cm and \(h = 12\) cm.
\(A_{trapezoid}=\frac{(16 + 24)\times12}{2}=\frac{40\times12}{2}=240\) \(cm^{2}\)

Step5: Find the area of the semicircle

The formula for the area of a semicircle is \(A_{semicircle}=\frac{1}{2}\pi r^{2}\). Substitute \(r = 8\) cm:
\(A_{semicircle}=\frac{1}{2}\pi\times8^{2}=\frac{1}{2}\pi\times64 = 32\pi\) \(cm^{2}\)

Step6: Find the area of the figure

The area of the figure is the area of the trapezoid minus the area of the semicircle.
\(A = A_{trapezoid}-A_{semicircle}=240-32\pi\)
If we take \(\pi\approx3.14\), then \(A=240 - 32\times3.14=240 - 100.48 = 139.52\) \(cm^{2}\)
(If we keep it in terms of \(\pi\), \(A=(240 - 32\pi)\) \(cm^{2}\))

Answer:

If we use \(\pi\approx3.14\), the area is \(139.52\space cm^{2}\) (or \(240 - 32\pi\space cm^{2}\))