A prism has a square base that measures 10 cm by 10 cm. A cylinder has a base with the same area as the base of the prism.
b) If both the prism and the cylinder are 20 cm high, which container will require less material to make?
c) Do both containers have the same volume? Explain.
b) The area of the base of the prism is 10 cm x 10 cm = 100 cm². Therefore, the area of the base of the cylinder must also be 100 cm². Since the base of a cylinder is a circle, we need to find the radius of the cylinder's base:
Area of a circle = πr²
100 cm² = πr²
r ≈ 5.64 cm
The height of both containers is 20 cm. To find the lateral area of the prism, we need to find the perimeter of the base and multiply by the height:
Perimeter of square base = 4 x 10 cm = 40 cm
Lateral area of prism = 40 cm x 20 cm = 800 cm²
To find the lateral area of the cylinder, we need to find the circumference of the base and multiply by the height:
Circumference of circle base = 2πr ≈ 35.44 cm
Lateral area of cylinder = 35.44 cm x 20 cm = 708.8 cm²
Therefore, the prism will require less material to make.
c) No, the prism and cylinder do not have the same volume. The volume of the prism is:
Volume of prism = base area x height = 100 cm² x 20 cm = 2000 cm³
The volume of the cylinder is:
Volume of cylinder = base area x height = πr² x 20 cm ≈ 1000 cm³
Therefore, the prism has twice the volume of the cylinder.