Question

a cylinder and a cone have the same base and altitude. if the volume of the cone is 30 cubic inches, what is the volume of the cylinder?
a. 10 cubic inches
b. 15 cubic inches
c. 30 cubic inches
d. 60 cubic inches
e. 90 cubic inches

Explanation:

Step1: Recall volume formulas

Volume of a cone: $V_{cone} = \frac{1}{3}\pi r^2 h$
Volume of a cylinder: $V_{cylinder} = \pi r^2 h$

Step2: Relate cone and cylinder volumes

Since they have the same base ($r$) and altitude ($h$), $V_{cylinder} = 3 \times V_{cone}$.

Step3: Calculate cylinder volume

Given $V_{cone} = 30$ cubic inches,
$V_{cylinder} = 3 \times 30 = 90$? Wait, no—wait, wait, no! Wait, wait, correction: Wait, no, wait, the formula: cone is 1/3 of cylinder. So cylinder is 3 times cone? Wait, no, wait, no—wait, no, the cone volume is (1/3)πr²h, cylinder is πr²h. So if cone is 30, then cylinder is 3*30? Wait, no, wait, no, that can't be. Wait, no, wait, maybe I mixed up. Wait, no, let's check again.

Wait, no, the problem: cone volume is 30. Cylinder with same base and height: $V_{cylinder} = 3 \times V_{cone}$? Wait, no, wait, no—wait, no, the cone is (1/3) of the cylinder. So cylinder is 3 times cone. So 3*30=90? But wait, the options have E as 90. Wait, but let me confirm.

Wait, volume of cone: $V_{cone} = \frac{1}{3}Bh$, where $B$ is base area. Volume of cylinder: $V_{cylinder} = Bh$. So if $V_{cone} = 30 = \frac{1}{3}Bh$, then $Bh = 90$. So $V_{cylinder} = Bh = 90$. So the answer is E.

Wait, but let me check the options again. The options are A:10, B:15, C:30, D:60, E:90. So yes, E is 90.

Answer:

E. 90 cubic inches