a circular cylinder has a base with a radius of 7.5 cm. and a height of 12 cm. a rectangular prism has a square base and height of 8.0 cm. if the cylinder and the prism have equal volumes, what is the length of the base of the prism to the nearest tenth?
The volumes are equal, so the equation below fits. Insert values and solve for L.
π*r^2*h = h*L^2
16.3 is the length of the base rounded to the nearest tenth.
3.14 x 7.5 x 7.5 x 12 = 2119.50
16.28 x 16.28 x 8 =2,120.3
To solve this problem, we first need to calculate the volume of the cylinder and the rectangle prism.
The volume of a circular cylinder is given by the formula:
V_cylinder = π * r^2 * h
where r is the radius of the base and h is the height of the cylinder.
Given that the radius of the cylinder base is 7.5 cm and the height is 12 cm, we can substitute these values into the formula to calculate the volume of the cylinder:
V_cylinder = π * (7.5 cm)^2 * 12 cm
V_cylinder = π * 56.25 cm^2 * 12 cm
V_cylinder = π * 675 cm^2
Now, let's calculate the volume of the rectangular prism.
The volume of a rectangular prism is given by the formula:
V_prism = l * w * h
where l is the length of the base, w is the width of the base, and h is the height of the prism.
Given that the prism has a square base, we can set the length and width of the base equal to each other. Let's call this length x.
V_prism = x * x * 8 cm
V_prism = 8x^2 cm^3
Since the volume of the cylinder and the prism are equal, we can set up the following equation:
π * 675 cm^2 = 8x^2 cm^3
To find the length of the base of the prism (x), we can rearrange the equation and solve for x:
8x^2 = π * 675
x^2 = (π * 675) / 8
x^2 = 84.375
x ≈ √84.375
x ≈ 9.2 cm (rounded to the nearest tenth)
Therefore, the length of the base of the prism is approximately 9.2 cm.