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 calculation of the volume of a sphere is when using \(\pi\) as opposed to using \(3.14\), we'll first need to calculate the volume of a sphere using both values for \(\pi\).

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

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

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

**1. Calculating volume using \(\pi\):**
- Given that the radius \(r = 39\) inches, the volume using \(\pi\) is:

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

Calculating \(39^3\):

\[
39^3 = 59319
\]

Then substituting back into the formula:

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

Using \(\pi \approx 3.14159\):

\[
V_\pi \approx 79192 \times 3.14159 \approx 248,159.6 \text{ cubic inches}
\]

**2. Calculating volume using \(3.14\):**
- Now using \(3.14\) instead of \(\pi\):

\[
V_{3.14} = \frac{4}{3} (3.14) (39)^3
\]
\[
V_{3.14} = \frac{4}{3} (3.14) (59319) = \frac{4 \times 3.14 \times 59319}{3}
\]

Calculating \(4 \times 3.14 = 12.56\):

Now, evaluate \(12.56 \times 59319\):

\[
12.56 \times 59319 \approx 745,736.44
\]

Now divide by 3:

\[
V_{3.14} = \frac{745736.44}{3} \approx 248578.81 \text{ cubic inches}
\]

**3. Finding the difference:**
Now, we find the difference in the volumes:

\[
\text{Difference} = V_\pi - V_{3.14} \approx 248159.6 - 248578.81 \approx -419.21
\]

This means the volume with \(3.14\) is actually larger than with \(\pi\). To find how much larger:

\[
\text{Difference} = 248578.81 - 248159.6 \approx 419.21
\]

**4. Rounding to the nearest tenth:**

The final calculation is:

\[
\text{Difference} \approx 419.2 \text{ cubic inches}
\]

So, the calculation of the volume of a sphere whose radius is 39 inches is approximately 419.2 cubic inches larger when using \(3.14\) instead of \(\pi\).