A partial cylinder lies on its side. The bases are a 90 degree sector of a circle. What is the exact volume of the partial cylinder. r=8,length 10
160 pi inch cubed
20 pi inch cubed
80 pi inch cubed
40 pi inch cubed
I will interpret this as having 1/4 of the cylinder
vol of whole cyl. = π(8^2)(10) = 640π
so 1/4 of that is 160π
To find the exact volume of the partial cylinder, we need to consider the formula for the volume of a cylinder.
The formula for the volume of a cylinder is V = πr^2h, where V is the volume, r is the radius of the base, and h is the height of the cylinder.
In this case, the radius of the base is given as 8 and the length of the cylinder is 10.
However, given that the bases are a 90-degree sector of a circle, we need to calculate the effective height of the cylinder.
Since the bases form a 90-degree sector, we can find the height by multiplying the radius by the angle in radians of the sector. In this case, the angle is 90 degrees or π/2 radians.
So, the height of the cylinder is h = 8 * (π/2) = 4π.
Now we can substitute the values of r and h into the volume formula:
V = π(8^2)(4π) = 64π * 4π = 256π^2.
Therefore, the exact volume of the partial cylinder is 256π^2 cubic units, which is approximately 804.25 cubic units.
Since the question asks for the answer in terms of pi, the correct option is 256π^2 inch cubed.
To find the volume of the partial cylinder, we need to calculate the volume of the entire cylinder and then subtract the volume of the missing portion.
First, let's find the volume of the entire cylinder. The formula for the volume of a cylinder is V = πr^2h, where r is the radius and h is the height (or length).
Given that the radius (r) is 8 and the length (h) is 10, we can calculate the volume of the entire cylinder:
V_entire = π(8^2)(10)
V_entire = π(64)(10)
V_entire = 640π cubic inches
Now, let's find the volume of the missing portion. The missing portion is a 90-degree sector of a circle, which means it is one-fourth of the total area of a circle with a radius of 8.
The formula for the area of a circle is A = πr^2, where r is the radius. Given that the radius (r) is 8, we can calculate the total area of the circle:
A_circle = π(8^2)
A_circle = π(64)
A_circle = 64π square inches
Since the missing portion is one-fourth of the total area, the area of the missing portion is:
A_missing = (1/4) * A_circle
A_missing = (1/4) * 64π
A_missing = 16π square inches
To find the volume of the missing portion, we multiply the area by the height (or length) of the cylinder:
V_missing = A_missing * h
V_missing = 16π * 10
V_missing = 160π cubic inches
Finally, to find the exact volume of the partial cylinder, we subtract the volume of the missing portion from the volume of the entire cylinder:
V_partial_cylinder = V_entire - V_missing
V_partial_cylinder = 640π - 160π
V_partial_cylinder = 480π cubic inches
So, the exact volume of the partial cylinder is 480π cubic inches. Therefore, the correct answer is 480π inch cubed.