The shaded region and the regions left blank in the given figures form
composite figures.
Correct responses:
6. 426.9 km²
7. 183 in.²
8. 192.42 m²
9. 239.4 ft.²
10. 89.22 cm²
Methods of calculation
Solution:
6. The area of a triangle = [tex]\mathbf{\frac{1}{2}}[/tex] × Base length × Height
Therefore;
Area of the triangle, A = [tex]\frac{1}{2}[/tex] × 40 km × 27 km = 540 km²
Area of a circle, A = [tex]\mathbf{\pi \cdot \frac{D^2}{4}}[/tex]
Where;
D = The diameter of the circle.
Area of the circle having a diameter of D = 12 km is; A = [tex]\pi \times \frac{12^2}{4}[/tex] ≈ 113.1 km²
Area of the shaded region, A = 540 km² - 113.1 km² = 426.9 km²
7. Taking the the shades region as two similar quadrilaterals, we have;
Length of side, x = √(13² + 9²) in. = 5·√(10) in.
Length of side, y = √(250 - 15²) = 5
[tex]Area \ of \ a \ trapezoid, \ A = \mathbf{ \dfrac{a + b}{2} \times h}[/tex]
[tex]Area \ of \ the \ trapezoid = \dfrac{5 + 14 + 26}{2} \times 15 = \mathbf{ 337.5}[/tex]
Sum of area of the two triangles left blank is therefore;
A = [tex]\frac{1}{2}[/tex] × 15 in. × 5 in. + [tex]\frac{1}{2}[/tex] × 26 in. × 9 in. = 154.5 in.²
Therefore;
- Area of the shade region, A = 337.5 in.² - 154.5 in.² = 183 in.²
8. Area of large circle = π × [tex]\frac{21^2}{4}[/tex] = 110.25·π
Area of the smaller circle left blank = π × [tex]\frac{\left(21 - 2 \times 3.5 \right)^2}{4}[/tex] = 49·π
Area of the shaded region = Area of the large circle - Area of the smaller circle left blank
Therefore;
- A = 110.25·π m² - 49·π m² ≈ 192.42 m²
9. Altitude of the cut out triangle = [tex]\sqrt{23.8^2 - 21^2}[/tex] = 11.2
Area of the triangle = [tex]\frac{1}{2}[/tex] × 21 ft. × 11.2 ft. = 117.6 ft.²
Area of a parallelogram = Base × Height
Therefore;
Area of the parallelogram = 23.8 ft. × 15 ft. = 357 ft.²
Area of the shaded region, A = 357 ft.² - 117.6 ft.² = 239.4 ft.²
10. Assumption; Type of quadrilateral = A square
Area of the square = 8 cm × 8 cm = 64 cm²
Diagonal of the square = √((8 cm)² + (8 cm)²) = 8·√2 cm
Diameter of the circle = The diagonal of the square
Area of the circle = π × [tex]\frac{\left(8 \cdot \sqrt{2} \ cm \right)^2 }{4}[/tex] = 32·π cm²
Area of the shaded region = 32·π cm² - 8·√2 cm² ≈ 89.22 cm²
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