Question

How much larger is the calculation of the volume of a sphere's whose radius is
39 inches if π is used instead of 3.14?
Round your answer to the nearest tenth.

Answer 1

To find out how much larger the volume of a sphere is when using \( \pi \) versus 3.14, we first need to calculate the volume using both values.

The formula for the volume \( V \) of a sphere is:

\[
V = \frac{4}{3} \pi r^3
\]

where \( r \) is the radius of the sphere.

### Step 1: Calculate the volume using \( \pi \)

1. Set \( r = 39 \) inches.
2. Using \( \pi \):

\[
V_{\pi} = \frac{4}{3} \pi (39)^3
\]

Calculating \( (39)^3 \):

\[
39^3 = 59319
\]

Thus,

\[
V_{\pi} = \frac{4}{3} \pi (59319) = \frac{4 \times 59319}{3} \pi = \frac{237276}{3} \pi = 79192 \pi
\]

Now, using the approximate value of \( \pi \approx 3.14159 \):

\[
V_{\pi} \approx 79192 \times 3.14159 \approx 248,800.4 \text{ cubic inches}
\]

### Step 2: Calculate the volume using \( 3.14 \)

Using \( r = 39 \) inches and \( \pi \approx 3.14 \):

\[
V_{3.14} = \frac{4}{3} (3.14) (39)^3
\]

This remains the same volume calculation as above for \( (39)^3 \):

\[
V_{3.14} = \frac{4}{3} (3.14) (59319)
\]

Calculating this:

\[
V_{3.14} \approx \frac{4 \times 3.14 \times 59319}{3}
\]

Calculating \( 4 \times 3.14 \times 59319 \):

\[
4 \times 3.14 = 12.56
\]

\[
12.56 \times 59319 \approx 745,133.64
\]

Now divide by 3:

\[
V_{3.14} \approx \frac{745133.64}{3} \approx 248,377.88 \text{ cubic inches}
\]

### Step 3: Find the difference

Now we subtract the two volumes to find out how much larger the volume calculated with \( \pi \) is compared to that calculated with \( 3.14 \):

\[
\text{Difference} = V_{\pi} - V_{3.14}
\]

\[
\text{Difference} \approx 248800.4 - 248377.88 \approx 422.52
\]

### Step 4: Round to the nearest tenth

Finally, rounding \( 422.52 \) to the nearest tenth gives us:

\[
\text{Difference} \approx 422.5 \text{ cubic inches}
\]

The volume of the sphere using \( \pi \) is approximately **422.5 cubic inches larger** than when using \( 3.14 \).