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A landscape architect is designing a garden in the shape of a composite figure. The garden consists of a rectangle ABCD with a semi-circle attached to one side, as shown in the diagram.
Given information:
– The total area of the garden is 180 m².
– AB = 12 m
– DE is perpendicular to BC and AD
– The area of triangle ADE is 24 m²
– The radius of the semi-circle is 4 m
Calculate the length of BC.
Let’s solve this step-by-step:
1. Let the length of BC be x meters.
2. Area of semi-circle = πr² / 2 = π * 4² / 2 = 8π m²
3. Area of rectangle ABCD = Total area – Area of semi-circle
180 – 8π = 12x
4. In triangle ADE:
Area = (1/2) * base * height
24 = (1/2) * 12 * DE
DE = 4 m
5. Area of rectangle BCDE = x * 4
6. Area of rectangle ABCD = Area of rectangle BCDE + Area of triangle ADE
12x = 4x + 24
7. Solving for x:
8x = 24
x = 3
8. However, this result doesn’t make sense with the given dimensions. Let’s recalculate:
Total area = Area of rectangle + Area of semi-circle
180 = 12x + 8π
180 – 8π = 12x
15 – 2π/3 = x
9. Calculating the value:
x ≈ 15 – 2.094 ≈ 12.906 m
Therefore, the correct length of BC is approximately 12.91 meters.
Verification:
– Area of rectangle: 12 * 12.906 ≈ 154.872 m²
– Area of semi-circle: 8π ≈ 25.133 m²
– Total area: 154.872 + 25.133 ≈ 180.005 m²
The total area is indeed very close to 180 m², confirming our corrected calculation.
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