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Difficulty: MediumArea of Two-Dimensional Shapes

A logo design consists of a region bounded by an isosceles trapezoid and a semicircle. The semicircle is attached to the shorter parallel side of the trapezoid such that the diameter of the semicircle is equal to the length of that side, and the semicircle lies entirely outside the trapezoid. The trapezoid has a height of 88 centimeters, a longer parallel side of 1818 centimeters, and a shorter parallel side of 1010 centimeters. What is the total area, in square centimeters, of the logo?

  1. A
    224+12.5π224 + 12.5\pi
  2. B
    112+50π112 + 50\pi
  3. 112+12.5π112 + 12.5\piAnswer
  4. D
    112+25π112 + 25\pi

Answer

The total area of the logo is 112+12.5π112 + 12.5\pi square centimeters.
The total area of the logo is the sum of the area of the trapezoid and the area of the semicircle. The area of the trapezoid is calculated using the formula A=b1+b22hA = \frac{b_1 + b_2}{2}h, which gives 18+102×8=112\frac{18 + 10}{2} \times 8 = 112 square centimeters. The area of the semicircle is half the area of a full circle with a diameter of 1010 centimeters (radius of 55 centimeters), which is 12πr2=12π(52)=12.5π\frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5^2) = 12.5\pi square centimeters. Adding these two areas together gives 112+12.5π112 + 12.5\pi square centimeters.

Step-by-Step Solution

1
Calculate the area of the trapezoid section of the logo.
The area of the trapezoid is 112112 square centimeters.
The area of a trapezoid is given by the formula A=b1+b22hA = \frac{b_1 + b_2}{2}h, where b1=18b_1 = 18, b2=10b_2 = 10, and h=8h = 8.
2
Calculate the area of the semicircle section of the logo.
The area of the semicircle is 12.5π12.5\pi square centimeters.
The diameter of the semicircle is the shorter base of the trapezoid (1010 centimeters), so the radius is r=5r = 5 centimeters. The area of a semicircle is half the area of a full circle: A=12πr2=12π(5)2=12.5πA = \frac{1}{2}\pi r^2 = \frac{1}{2}\pi (5)^2 = 12.5\pi.
3
Sum the areas of the trapezoid and the semicircle to find the total area.
The total area is 112+12.5π112 + 12.5\pi square centimeters.
The total area of a composite figure is the sum of the areas of its non-overlapping component parts.

Key Concept

Calculating the total area of a composite figure by decomposing it into standard shapes (a trapezoid and a semicircle) and summing their individual areas.
Estimated Time:1m 30s
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Topics

Area of Two-Dimensional Shapes

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