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 \(\pi\).