Here are the step-by-step solutions for each problem. To find the volume of prisms and cylinders, we generally use the formula:
Volume = Area of Base × Height
1) Cylinder
*
Identify the shape: It is a cylinder.
*
Find the area of the base: The base is a circle with a radius ($r$) of 8 km.
* Area = $\pi \times r^2$
* Area = $\pi \times 8^2 = 64\pi$
*
Calculate Volume: Multiply the base area by the height ($h = 7$ km).
* Volume = $64\pi \times 7 = 448\pi$
* Using $\pi \approx 3.14159$: $448 \times 3.14159 \approx 1407.43$
*
Round to nearest tenth: 1407.4
2) Hexagonal Prism
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Identify the shape: A prism with a regular hexagon base.
*
Find the area of the base: For a regular hexagon, Area = $\frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$.
* Side length = 3 ft, so Perimeter = $6 \times 3 = 18$ ft.
* Apothem = 2.6 ft.
* Base Area = $0.5 \times 18 \times 2.6 = 23.4$ sq ft.
*
Calculate Volume: Multiply by the height ($h = 4$ ft).
* Volume = $23.4 \times 4 = 93.6$
3) Rectangular Prism
*
Identify the shape: A box shape (rectangular prism).
*
Formula: Volume = Length × Width × Height.
*
Calculation:
* Volume = $9 \text{ cm} \times 6 \text{ cm} \times 2 \text{ cm}$
* Volume = $54 \times 2 = 108$
4) Pentagonal Prism
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Identify the shape: A prism with a regular pentagon base.
*
Find the area of the base: Area = $\frac{1}{2} \times \text{Perimeter} \times \text{Apothem}$.
* Side length = 5.5 m, so Perimeter = $5 \times 5.5 = 27.5$ m.
* Apothem = 3.8 m.
* Base Area = $0.5 \times 27.5 \times 3.8 = 52.25$ sq m.
*
Calculate Volume: Multiply by the height ($h = 8$ m).
* Volume = $52.25 \times 8 = 418$
5) Trapezoidal Prism
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Identify the shape: A prism where the base is a trapezoid.
*
Find the area of the base (trapezoid): Area = $\frac{1}{2} \times (\text{base}_1 + \text{base}_2) \times \text{height}_{\text{trap}}$.
* Bases are 2 ft and 4 ft. The height of the trapezoid itself is 3 ft.
* Base Area = $0.5 \times (2 + 4) \times 3$
* Base Area = $0.5 \times 6 \times 3 = 9$ sq ft.
*
Calculate Volume: Multiply by the length/height of the prism ($h = 5$ ft).
* Volume = $9 \times 5 = 45$
6) Cylinder
*
Identify the shape: It is a cylinder.
*
Find the area of the base: The base is a circle with a radius ($r$) of 5 in.
* Area = $\pi \times r^2$
* Area = $\pi \times 5^2 = 25\pi$
*
Calculate Volume: Multiply by the height ($h = 4$ in).
* Volume = $25\pi \times 4 = 100\pi$
* Using $\pi \approx 3.14159$: $100 \times 3.14159 \approx 314.159$
*
Round to nearest tenth: 314.2
Final Answer:
1) 1407.4 km³
2) 93.6 ft³
3) 108 cm³
4) 418 m³
5) 45 ft³
6) 314.2 in³
Parent Tip: Review the logic above to help your child master the concept of volume of prism and cylinder worksheet.
