Question

what is the volume of this cylinder? round your answer to the nearest hundredth. image of a cylinder with radius 13 mm and height 20 mm options: 10,613.20; 5306.60; 2653.30; 1326.65. what is the volume of this cylinder? round your answer to the nearest hundredth. image of a cylinder with radius 16 cm and height 18 cm options: 14,469.12; 28938.24; 9646.08; 7234.56.

Explanation:

Step1: Recall the formula for the volume of a cylinder

The formula for the volume \( V \) of a cylinder is \( V=\pi r^{2}h \), where \( r \) is the radius of the base and \( h \) is the height of the cylinder.

Step2: Solve for the first cylinder (with radius \( r = 13\space mm \) and height \( h=20\space mm \))

Substitute \( r = 13 \) and \( h = 20 \) into the formula:

$$
LATEXBLOCK0
$$

Wait, no, wait. Wait, maybe I made a mistake. Wait, let's check again. Wait, the radius is 13 mm, height is 20 mm. Wait, \( \pi r^{2}h=\pi\times13^{2}\times20=\pi\times169\times20 = 3380\pi\approx3380\times3.14159265\approx10613.20 \). But the options include 10613.20, 5306.60, etc. Wait, maybe the radius is the diameter? Wait, the diagram shows 13 mm as the radius? Wait, no, maybe the 13 mm is the diameter? Wait, the problem says "13 mm" with an arrow to the radius? Wait, the diagram: the arrow is on the radius? Wait, maybe I misread. Wait, if the 13 mm is the radius, then the calculation is as above. But if it's the diameter, then radius is \( 13/2 = 6.5 \) mm. Let's check that. If \( r = 6.5 \), \( h = 20 \):

$$
LATEXBLOCK1
$$

Wait, that's one of the options (2653.30). Wait, maybe the 13 mm is the diameter? Wait, the diagram: the arrow is on the radius? Wait, maybe the problem has a typo, or I misinterpret the diagram. Wait, let's check the options. The options are 10613.20, 5306.60, 2653.30, 1326.65. Let's recalculate. If radius is 13, then \( V=\pi\times13^{2}\times20 = 3380\pi\approx10613.20 \). If radius is 6.5 (diameter 13), then \( V=\pi\times6.5^{2}\times20=845\pi\approx2653.30 \). Let's check the second cylinder. For the second cylinder, radius is 16 cm, height 18 cm. If radius is 16, then \( V=\pi\times16^{2}\times18=\pi\times256\times18 = 4608\pi\approx14469.12 \), which is one of the options (14469.12). So maybe in the first problem, the 13 mm is the diameter, so radius is 6.5 mm. Let's confirm.

Wait, the first cylinder: if the 13 mm is the diameter, then radius \( r = 13/2 = 6.5 \) mm, height \( h = 20 \) mm. Then volume \( V=\pi r^{2}h=\pi\times(6.5)^{2}\times20=\pi\times42.25\times20 = 845\pi\approx845\times3.14159265\approx2653.30 \), which is option C (2653.30).

For the second cylinder: radius \( r = 16 \) cm, height \( h = 18 \) cm. Then volume \( V=\pi r^{2}h=\pi\times16^{2}\times18=\pi\times256\times18 = 4608\pi\approx4608\times3.14159265\approx14469.12 \), which is option A (14469.12).

Wait, so maybe in the first problem, the 13 mm is the diameter, so radius is 6.5 mm. Let's redo the first cylinder:

First cylinder:

• Diameter \( d = 13 \) mm, so radius \( r=\frac{d}{2}=\frac{13}{2} = 6.5 \) mm.
• Height \( h = 20 \) mm.
• Volume \( V=\pi r^{2}h=\pi\times(6.5)^{2}\times20 \).
• Calculate \( (6.5)^{2}=42.25 \).
• Then \( 42.25\times20 = 845 \).
• Then \( 845\times\pi\approx845\times3.14159265\approx2653.30 \). So the answer for the first cylinder is 2653.30.

Second cylinder:

• Radius \( r = 16 \) cm.
• Height \( h = 18 \) cm.
• Volume \( V=\pi r^{2}h=\pi\times(16)^{2}\times18 \).
• Calculate \( 16^{2}=256 \).
• Then \( 256\times18 = 4608 \).
• Then \( 4608\times\pi\approx4608\times3.14159265\approx14469.12 \). So the answer for the second cylinder is 14469.12.

Answer:

First cylinder: 2653.30 (Option: 2653.30)
Second cylinder: 14469.12 (Option: 14469.12)