Answer: B) 128π ft³
Step-by-step explanation: To find the volume of a cylinder, we need to use the formula:
\ce {V = πr^2h} V = πr^2h
where \ce {V} V is the volume, \ce {π} π is a constant that is approximately equal to 3.14, \ce {r} r is the radius of the base of the cylinder, and \ce {h} h is the height of the cylinder. In this problem, we are given that the radius is 4 ft and the height is 8 ft. Plugging these values into the formula, we get:
\ce {V = π(4)^2(8)} V = π(4)^2(8)
\ce {V = π(16)(8)} V = π(16)(8)
\ce {V = 128π} V = 128π
Therefore, the exact volume of the cylinder is 128π ft³. This is option B in the choices given. Note that if we want to find the approximate volume, we can use a calculator to multiply 128 by 3.14 and get 401.92 ft³. However, this is not an exact answer because it involves rounding off π to two decimal places.
Here is why the other options are incorrect (for further clarification).
A) 32π ft³
This option is incorrect because it is obtained by using the formula for the volume of a cone, not a cylinder. The formula for the volume of a cone is:
\ce {V = (1/3)πr^2h} V = (1/3)πr^2h
If we use this formula with the given values of r and h, we get:
\ce {V = (1/3)π(4)^2(8)} V = (1/3)π(4)^2(8)
\ce {V = (1/3)π(16)(8)} V = (1/3)π(16)(8)
\ce {V = 32π} V = 32π
However, this is not the correct formula for a cylinder, so this option is wrong.
C) 256π ft³
This option is incorrect because it is obtained by using the formula for the volume of a sphere, not a cylinder. The formula for the volume of a sphere is:
\ce {V = (4/3)πr^3} V = (4/3)πr^3
If we use this formula with the given value of r, we get:
\ce {V = (4/3)π(4)^3} V = (4/3)π(4)^3
\ce {V = (4/3)π(64)} V = (4/3)π(64)
\ce {V = 256π} V = 256π
However, this is not the correct formula for a cylinder, and it does not use the given value of h, so this option is wrong.
D) 803.84π ft³
This option is incorrect because it is obtained by using an approximate value of π, not the exact value. If we use 3.14 as an approximation for π and multiply it by 128, we get:
\ce {V = 128(3.14)} V = 128(3.14)
\ce {V = 401.92} V = 401.92
However, this is not an exact answer because it involves rounding off π to two decimal places. The exact answer is 128π ft³, which is option B.