Answer:
The radius of the planter, rounded to the nearest inch is, 4 inches
Step-by-step 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:
[tex]V = \frac{2}{3} \pi r^3[/tex] ....[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 [tex]\frac{3}{2}[/tex] we have;
[tex]\frac{3}{2}V = \pi r^3[/tex]
Divide both sides by [tex]\pi[/tex] we have;
[tex]\frac{3 V}{2 \pi} =r^3[/tex]
or
[tex]r^3 = \frac{3V}{2 \pi}[/tex]
⇒[tex]r = \sqrt[3]{\frac{3V}{2 \pi}}[/tex] ....[2]
It is also given that:
The label on the planter says that it holds approximately 134 cubic inches of potting soil.
⇒[tex]V = 134 in^3[/tex] and use 3.14 for pi.
Substitute these in [2] we have;
[tex]r = \sqrt[3]{\frac{3 \cdot 134}{2 \cdot 3.14}}[/tex]
⇒[tex]r = \sqrt[3]{\frac{402}{6.28}}= \sqrt[3]{64.0127389}[/tex]
Simplify:
r = 4.00026538 inches
Therefore, the radius of the planter, rounded to the nearest inch is, 4 inches
