Question

A rectangle prism has a hemisphere cut out of it to make a mold. The hemisphere's diameter is 2.5 inches. The rectangular prism has a length of 6 inches, a width of 6.5 inches and a height of 8 inches. Answer the following questions. Use pi and round to the nearest tenth if necessary. Show all work.

1. What is the volume of the prism

2. What is the volume of the hemisphere

3.What is the volume of the entire solid

Answer 1

To solve these questions, we'll need to calculate the volume of the rectangular prism, the volume of the hemisphere, and then subtract the volume of the hemisphere from the volume of the rectangular prism.

### 1. Volume of the Prism

The volume \( V \) of a rectangular prism can be calculated with the formula:

\[
V = l \times w \times h
\]

where:
- \( l \) = length
- \( w \) = width
- \( h \) = height

Given:
- Length \( l = 6 \) inches
- Width \( w = 6.5 \) inches
- Height \( h = 8 \) inches

Substituting the values:

\[
V = 6 \times 6.5 \times 8
\]

Calculating:

\[
6 \times 6.5 = 39
\]
\[
39 \times 8 = 312
\]

So, the volume of the rectangular prism is:

\[
\text{Volume of the prism} = 312 \text{ cubic inches}
\]

### 2. Volume of the Hemisphere

The volume \( V \) of a hemisphere can be calculated with the formula:

\[
V = \frac{2}{3} \pi r^3
\]

where \( r \) is the radius of the hemisphere. The diameter of the hemisphere is given as 2.5 inches, so the radius \( r \) is:

\[
r = \frac{2.5}{2} = 1.25 \text{ inches}
\]

Now substituting the radius into the formula:

\[
V = \frac{2}{3} \pi (1.25)^3
\]

Calculating \( (1.25)^3 \):

\[
(1.25)^3 = 1.25 \times 1.25 \times 1.25
\]
Calculating \( 1.25 \times 1.25 = 1.5625 \)
Then:
\[
1.5625 \times 1.25 = 1.953125
\]

Now substituting back into the volume formula:

\[
V = \frac{2}{3} \pi (1.953125)
\]

Now, we approximate \( \pi \approx 3.14 \):

\[
V \approx \frac{2}{3} \times 3.14 \times 1.953125
\]

Calculating \( 3.14 \times 1.953125 = 6.1309375 \)

Now:

\[
V \approx \frac{2}{3} \times 6.1309375 \approx 4.08729167
\]

Finally multiplying by \( \frac{2}{3} \):

\[
V \approx 2.725
\]

Rounding to the nearest tenth:

\[
\text{Volume of the hemisphere} \approx 2.7 \text{ cubic inches}
\]

### 3. Volume of the Entire Solid

To calculate the volume of the entire solid, we start with the volume of the prism and subtract the volume of the hemisphere that is cut out:

\[
\text{Volume of entire solid} = \text{Volume of prism} - \text{Volume of hemisphere}
\]
\[
= 312 - 2.7
\]
\[
= 309.3
\]

### Final Answers:

1. Volume of the prism: \( \boxed{312} \) cubic inches
2. Volume of the hemisphere: \( \boxed{2.7} \) cubic inches
3. Volume of the entire solid: \( \boxed{309.3} \) cubic inches