Question

A gardener purchases a ceramic planter, in the shape of a hemisphere, for a small batch of leftover annuals. The volume of a hemisphere is modeled by the function V = (2/3) Πr ^3

Write a model for the radius as a function of the volume.
The label on the planter says that it holds approximately 134 cubic inches of potting soil. What is the radius of the planter, rounded to the nearest inch? Use 3.14 for pi

Answer 1

`V=\frac23\pi r^3`

`\frac3{2\pi}V=r^3`

`r=\sqrt[3]{\frac3{2\pi}V}`

Given that the volume of the planter is 134 cubic in., the radius is approximately


`r=\sqrt[3]{\frac3{2*3.14}*134}=1.8568\approx2\text{ in}`

Answer 2

The radius of the planter, rounded to the nearest inch is, 4 inches

Explanation:

As per the statement:

A gardener purchases a ceramic planter, in the shape of a hemisphere, for a small batch of leftover annuals

The volume of a hemisphere is modeled by the function as:

`V = (2)/(3) \pi r^3` ....[1]

where,

V is the volume and r is the radius of the hemisphere.

To write a model for the radius as a function of the volume.

Divide equation [1] to both sides by
`(3)/(2)` we have;

`(3)/(2)V = \pi r^3`

Divide both sides by
`\pi` we have;

`(3 V)/(2 \pi) =r^3`

or

`r^3 = (3V)/(2 \pi)`

⇒
`r = \sqrt[3]{(3V)/(2 \pi)}` ....[2]

It is also given that:

The label on the planter says that it holds approximately 134 cubic inches of potting soil.

⇒
`V = 134 in^3` and use 3.14 for pi.

Substitute these in [2] we have;

`r = \sqrt[3]{(3 \cdot 134)/(2 \cdot 3.14)}`

⇒
`r = \sqrt[3]{(402)/(6.28)}= \sqrt[3]{64.0127389}`

Simplify:

r = 4.00026538 inches

Therefore, the radius of the planter, rounded to the nearest inch is, 4 inches