Volume of Spheres Quick Check

Let's solve the problems step by step.

1. **Volume of a basketball with diameter 9.2 inches:**

The formula for the volume of a sphere is:
\[
V = \frac{4}{3} \pi r^3
\]
First, find the radius (r), which is half of the diameter:
\[
r = \frac{9.2}{2} = 4.6 \text{ inches}
\]
Plug in the values and use 3.14 for \(\pi\):
\[
V = \frac{4}{3} \times 3.14 \times (4.6)^3
\]
Computing the volume:
\[
V = \frac{4}{3} \times 3.14 \times 97.336
\]
\[
V \approx 407.51 \text{ in}^3
\]
So, the correct answer is:
\[
407.51 \text{ inches cubed}
\]

2. **Volume of a sphere with radius 1.8 yards:**

Using the formula \( V = \frac{4}{3} \pi r^3 \):
\[
V = \frac{4}{3} \pi (1.8)^3
\]
Let’s compute \( (1.8)^3 \):
\[
(1.8)^3 = 5.832
\]
So,
\[
V = \frac{4}{3} \pi \times 5.832
\]
\[
V \approx 7.776 \pi \, \text{yds}^3
\]
Rounded to the nearest hundredth:
\[
V \approx 7.78 \pi \, \text{yds}^3
\]
The correct answer is:
\[
7.78 \pi \text{ yds}^3
\]

3. **Air needed to fill a basketball with diameter 9 inches:**

Radius \( r = \frac{9}{2} = 4.5 \text{ inches} \):
\[
V = \frac{4}{3} \times 3.14 \times (4.5)^3
\]
Compute \( (4.5)^3 \):
\[
(4.5)^3 = 91.125
\]
\[
V = \frac{4}{3} \times 3.14 \times 91.125
\]
\[
V \approx 381.51 \text{ in}^3
\]
Rounded to the nearest tenth:
\[
V \approx 381.5 \text{ in}^3
\]
The correct answer is:
\[
381.5 \text{ inches cubed}
\]

4 and 5. **Unused space in the container with total volume 4000 mm³ and 50 marbles with radius 2 mm:**
Calculate the volume of one marble:
\[
r = 2 \text{ mm}
\]
Using the volume formula for a sphere:
\[
V = \frac{4}{3} \times 3.14 \times (2)^3
\]
Compute:
\[
V = \frac{4}{3} \times 3.14 \times 8
\]
\[
V = \frac{4}{3} \times 25.12
\]
\[
V \approx 33.49 \text{ mm}^3
\]
Volume of 50 marbles:
\[
50 \times 33.49 \approx 1674.5 \text{ mm}^3
\]
Unused space can be calculated as:
\[
4000 \text{ mm}^3 - 1674.5 \text{ mm}^3 = 2325.5 \text{ mm}^3
\]
Round to the nearest tenth:
\[
2325.5 \approx 2325.3
\]
The correct answer is:
\[
2,325.3 \text{ mm}^3
\]

So, the correct answers are:
1. 407.51 inches cubed
2. 7.78 π yds cubed
3. 381.5 inches cubed
4. 2,325.3 mm cubed
5. 2,325.3 mm cubed