Here are the step-by-step solutions for each problem on the worksheet.
1) Cone
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Formula: Volume = $\frac{1}{3} \times \pi \times r^2 \times h$
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Given: Radius ($r$) = 14 ft, Height ($h$) = 18 ft.
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Calculation:
* $r^2 = 14 \times 14 = 196$
* Volume = $\frac{1}{3} \times \pi \times 196 \times 18$
* Simplify: $\frac{1}{3} \times 18 = 6$
* Volume = $196 \times 6 \times \pi$
* Volume = $1176\pi$ cubic feet.
* (Approximate value: $1176 \times 3.14159 \approx 3694.51$)
2) Sphere
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Formula: Volume = $\frac{4}{3} \times \pi \times r^3$
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Given: Diameter = 18 in, so Radius ($r$) = 9 in.
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Calculation:
* $r^3 = 9 \times 9 \times 9 = 729$
* Volume = $\frac{4}{3} \times \pi \times 729$
* Simplify: $729 \div 3 = 243$
* Volume = $4 \times 243 \times \pi$
* Volume = $972\pi$ cubic inches.
* (Approximate value: $972 \times 3.14159 \approx 3053.63$)
3) Rectangular Prism
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Formula: Volume = Length $\times$ Width $\times$ Height
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Given: 10 yd, 2 yd, 5 yd.
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Calculation:
* Volume = $10 \times 2 \times 5$
* Volume = $100$ cubic yards.
4) Triangular Prism
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Formula: Volume = Area of Base Triangle $\times$ Length
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Area of Triangle: $\frac{1}{2} \times \text{base} \times \text{height}$
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Given: Triangle base = 12 yd, Triangle height = 8 yd, Prism length = 20 yd.
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Calculation:
* Area of Triangle = $\frac{1}{2} \times 12 \times 8 = 48$ sq yd.
* Volume = $48 \times 20$
* Volume = $960$ cubic yards.
5) Rectangular Prism
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Formula: Volume = Length $\times$ Width $\times$ Height
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Given: 6 ft, 3 ft, 8 ft.
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Calculation:
* Volume = $6 \times 3 \times 8$
* Volume = $18 \times 8$
* Volume = $144$ cubic feet.
6) Cylinder
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Formula: Volume = $\pi \times r^2 \times h$
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Given: Diameter = 8 in, so Radius ($r$) = 4 in. Height ($h$) = 8 in.
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Calculation:
* $r^2 = 4 \times 4 = 16$
* Volume = $\pi \times 16 \times 8$
* Volume = $128\pi$ cubic inches.
* (Approximate value: $128 \times 3.14159 \approx 402.12$)
7) Hemisphere (Half a Sphere)
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Formula: Volume = $\frac{1}{2} \times (\frac{4}{3} \times \pi \times r^3)$ which simplifies to $\frac{2}{3} \times \pi \times r^3$
*
Given: Radius ($r$) = 10 in.
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Calculation:
* $r^3 = 10 \times 10 \times 10 = 1000$
* Volume = $\frac{2}{3} \times \pi \times 1000$
* Volume = $\frac{2000}{3}\pi$ or $666.67\pi$ cubic inches.
* (Approximate value: $666.67 \times 3.14159 \approx 2094.40$)
8) Rectangular Prism
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Formula: Volume = Length $\times$ Width $\times$ Height
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Given: 10 yd, 5 yd, 5 yd.
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Calculation:
* Volume = $10 \times 5 \times 5$
* Volume = $50 \times 5$
* Volume = $250$ cubic yards.
9) Cone
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Formula: Volume = $\frac{1}{3} \times \pi \times r^2 \times h$
*
Given: Radius ($r$) = 8 ft, Height ($h$) = 18 ft.
*
Calculation:
* $r^2 = 8 \times 8 = 64$
* Volume = $\frac{1}{3} \times \pi \times 64 \times 18$
* Simplify: $\frac{1}{3} \times 18 = 6$
* Volume = $64 \times 6 \times \pi$
* Volume = $384\pi$ cubic feet.
* (Approximate value: $384 \times 3.14159 \approx 1206.37$)
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Final Answer:
1) $1176\pi \
Parent Tip: Review the logic above to help your child master the concept of area mixed shapes worksheet.
