Certainly! Let's tackle the problem step-by-step.
### Step 1: Write a model for the radius as a function of the volume.
We start with the volume equation of a hemisphere:
[tex]\[ V = \frac{2}{3} \pi r^3 \][/tex]
We need to solve this equation for [tex]\( r \)[/tex].
First, isolate [tex]\( r^3 \)[/tex] by multiplying both sides of the equation by [tex]\( \frac{3}{2\pi} \)[/tex]:
[tex]\[ V \cdot \frac{3}{2\pi} = r^3 \][/tex]
Now, take the cube root of both sides to solve for [tex]\( r \)[/tex]:
[tex]\[ r = \left( \frac{3V}{2\pi} \right)^{\frac{1}{3}} \][/tex]
This gives us the model for the radius as a function of the volume:
[tex]\[ r = \left( \frac{3V}{2\pi} \right)^{\frac{1}{3}} \][/tex]
### Step 2: Calculate the radius of the planter
Given:
- Volume [tex]\( V = 134 \)[/tex] cubic inches
- Use [tex]\( \pi = 3.14 \)[/tex]
Plug the given values into the radius model:
[tex]\[ r = \left( \frac{3 \cdot 134}{2 \cdot 3.14} \right)^{\frac{1}{3}} \][/tex]
First, calculate the fraction inside the parentheses:
[tex]\[ \frac{3 \cdot 134}{2 \cdot 3.14} = \frac{402}{6.28} \approx 63.98 \][/tex]
Next, take the cube root of this value:
[tex]\[ r = 63.98^{\frac{1}{3}} \approx 4.000265 \][/tex]
### Rounding to the nearest inch:
The radius, when rounded to the nearest inch, is:
[tex]\[ r \approx 4 \][/tex]
Therefore, the radius of the planter is approximately 4 inches.
