Question

A cylinder has a volume of 48π cm3 and height h. Complete this table for volume of cylinders with the same radius but different heights.

height (cm) volume (cm3)
h 48π
2h ? answer in pi
5h ? answer in pi
h/2 ? answer in pi
h/5 ? answer in pi

Answer 1

To calculate the volume for different heights, the volume formula V = πr²h is used, showing that the volume changes proportionally with height. The completed table includes volumes for heights 2h, 5h, h/2, and h/5.

Step-by-step explanation:

The volume of a cylinder is given by the formula V = πr²h, where r is the radius, h is the height, and π (pi) represents the mathematical constant roughly equal to 3.14159. Given the volume of the cylinder is 48π cm³ for a height h, changing the height will proportionately change the volume. We can complete the table as follows:

For height 2h: Volume = πr²(2h) = 2(πr²h) = 2(48π cm³) = 96π cm³

For height 5h: Volume = πr²(5h) = 5(πr²h) = 5(48π cm³) = 240π cm³

For height h/2: Volume = πr²(h/2) = 1/2(πr²h) = 1/2(48π cm³) = 24π cm³

For height h/5: Volume = πr²(h/5) = 1/5(πr²h) = 1/5(48π cm³) = 9.6π cm³

Answer 2

(i) For
`2\cdot h`, the volume is
`96\pi` cubic centimeters.

(ii) For
`5\cdot h`, the volume is
`240\pi` cubic centimeters.

(iii) For
`(h)/(2)`, the volume is
`24\pi` cubic centimeters.

(iv) For
`(h)/(5)`, the volume is
`9.6\pi` cubic centimeters.

Step-by-step explanation:

The volume of the cylinder (
`V`), measured in cubic centimeters, is defined by the following formula:

`V = \pi\cdot r^(2)\cdot h` (1)

Where:

`r` - Radius, measured in centimeters.

`h` - Height, measured in centimeters.

From statement, we understand that volume of the cylinder is directly proportional to its height. That is:

`V \propto h`

`V = k\cdot h` (2)

Where
`k` is the proportionality constant, measured in square centimeters.

In addition, we eliminate this constant by constructing the following relationship:

`(V_(2))/(V_(1)) = (h_(2))/(h_(1))`

`V_(2) = (h_(2))/(h_(1)) \cdot V_(1)` (3)

Based on (3) and knowing that
`V_(1) = 48\pi`, we calculate the volumes for each height ratio:

(i) For
`2\cdot h`, the volume is
`96\pi` cubic centimeters.

(ii) For
`5\cdot h`, the volume is
`240\pi` cubic centimeters.

(iii) For
`(h)/(2)`, the volume is
`24\pi` cubic centimeters.

(iv) For
`(h)/(5)`, the volume is
`9.6\pi` cubic centimeters.