Question

The volume of a cylinder can be determined using the formula V=πr2h, where r and h represent the radius and height of the cylinder, respectively. A volume of paint expressed as (8x3 + 31x2 + 32x)π and a volume of paint expressed as (10x3 + 17x2)π are poured into a paint can in the shape of a cylinder. Determine possible expressions for the radius of the can and the depth of the paint in the can.

Answer 1

Possible expressions for the radius of the can and the depth of the paint in the can are
`r = \sqrt{9\cdot x^(2)+24\cdot x+16}` and
`h = 2\cdot x`, respectively.

Explanation:

Let be the initial volumes of the initial cans represented by these expressions:

`V_(1) = (8\cdot x^(3)+31\cdot x^(2)+32\cdot x)\cdot \pi` (1)

`V_(2) = (10\cdot x^(3)+17\cdot x^(2))\cdot \pi` (2)

The resulting volume of the paint can is the sum of the two functions:

`V_(3) = (18\cdot x^(3)+48\cdot x^(2)+32\cdot x)\cdot \pi` (3)

Then, we proceed to factor the polynomial:

`V_(3) = 2\cdot (9\cdot x^(2)+24\cdot x +16)\cdot x \cdot \pi`

`V_(3) = \pi\cdot (9\cdot x^(2)+24\cdot x + 16)\cdot (2\cdot x)` (3b)

By direct comparison with the volume formula for the cylinder we have the following expressions:

`r^(2) = 9\cdot x^(2)+24\cdot x + 16`

`r = \sqrt{9\cdot x^(2)+24\cdot x+16}`

`h = 2\cdot x`

Possible expressions for the radius of the can and the depth of the paint in the can are
`r = \sqrt{9\cdot x^(2)+24\cdot x+16}` and
`h = 2\cdot x`, respectively.