Question

1. the volume of a cone is 36π cubic units. what is the volume of a cylinder with the same radius and the same height?
2. the volume of a cylinder is 175π cubic units. what is the volume of a cone with the same radius and the same height?
3. a cylinder and a cone have the same height and radius. the height of each is 5 centimeters, and the radius is 2 centimeters. calculate the volume of the cylinder and the cone (rounded to the nearest tenth). use 3.14 as an approximation for π.

cylinder:
cone:
problems 4–6: this table shows the radiuses of four cones with a height of 18 meters.

1. complete the table with the volume of each cone.
2. based on your table, if the radius of a cone doubles, does the volume also double? explain your thinking.
3. based on your table, is the relationship between the radius of a cone and its volume linear? explain your thinking.

(charts and images: cone and cylinder diagrams for 1,2; table for 4 - 6 with radius (m) 1,2,3,4 and volume (cu. m) columns)

Explanation:

Step1: Relate cone and cylinder volumes

The volume of a cone is $\frac{1}{3}$ the volume of a cylinder with the same radius and height: $V_{cone}=\frac{1}{3}V_{cylinder}$, so $V_{cylinder}=3V_{cone}$.

Step2: Solve Q1: Find cylinder volume

Substitute $V_{cone}=36\pi$:
$V_{cylinder}=3\times36\pi=108\pi$

Step3: Solve Q2: Find cone volume

Use $V_{cone}=\frac{1}{3}V_{cylinder}$, substitute $V_{cylinder}=175\pi$:
$V_{cone}=\frac{1}{3}\times175\pi=\frac{175\pi}{3}$

Step4: Solve Q3: Cylinder volume

Use $V_{cylinder}=\pi r^2 h$, substitute $r=2, h=5, \pi=3.14$:
$V_{cylinder}=3.14\times2^2\times5=3.14\times4\times5=62.8$

Step5: Solve Q3: Cone volume

Use $V_{cone}=\frac{1}{3}\pi r^2 h$, substitute values:
$V_{cone}=\frac{1}{3}\times3.14\times2^2\times5=\frac{1}{3}\times62.8\approx20.9$

Step6: Solve Q4: Calculate cone volumes

Use $V_{cone}=\frac{1}{3}\pi r^2 h$, $h=18$, $\pi=3.14$:

• $r=1$: $V=\frac{1}{3}\times3.14\times1^2\times18=18.84$
• $r=2$: $V=\frac{1}{3}\times3.14\times2^2\times18=75.36$
• $r=3$: $V=\frac{1}{3}\times3.14\times3^2\times18=169.56$
• $r=4$: $V=\frac{1}{3}\times3.14\times4^2\times18=301.44$

Step7: Answer Q5: Radius double check

Volume depends on $r^2$. When $r$ doubles, $r^2$ becomes $4$ times, so volume is 4x, not 2x.

Step8: Answer Q6: Linear relationship check

Linear relationships have constant rate of change. Here volume changes by multiples of $r^2$, not constant, so not linear.

Answer:

1. $108\pi$ cubic units
2. $\frac{175\pi}{3}$ cubic units
3. Cylinder: $62.8$ cubic centimeters; Cone: $20.9$ cubic centimeters

4.

Radius (m)	Volume (cu. m)
2	75.36
3	169.56
4	301.44

1. No, the volume does not double. The volume of a cone is proportional to the square of its radius, so when the radius doubles, the volume becomes $2^2=4$ times the original volume, not 2 times.
2. No, the relationship is not linear. A linear relationship has a constant rate of change, but the volume of the cone increases with the square of the radius, so the rate of change of volume relative to radius increases as radius increases.