To find the volume of the globe, we can use the formula for the volume of a sphere, which is given by:
\[ V = \frac{4}{3} \pi r^3 \]
where \( r \) is the radius of the sphere.
-
Calculate the radius of the original globe:
The diameter of the original globe is 20 inches, so the radius \( r \) is:
\[ r = \frac{20 \text{ inches}}{2} = 10 \text{ inches} \]
-
Calculate the volume of the original globe:
Substituting the radius into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times (10)^3 \]
First, calculate \( (10)^3 \):
\[ (10)^3 = 1000 \]
Now substitute it into the volume formula:
\[ V = \frac{4}{3} \times 3.14 \times 1000 \]
Calculate \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \approx 1.3333 \quad \Rightarrow \quad 1.3333 \times 3.14 \approx 4.1888 \]
Now multiply by 1000:
\[ V \approx 4.1888 \times 1000 \approx 4188.8 \text{ in}^3 \]
-
Reduce the dimensions by half:
If the dimensions are reduced by half, the new diameter will be:
\[ \text{New Diameter} = \frac{20}{2} = 10 \text{ inches} \]
Then the new radius \( r' \) will be:
\[ r' = \frac{10}{2} = 5 \text{ inches} \]
-
Calculate the volume of the smaller globe:
Now substituting the new radius into the volume formula:
\[ V' = \frac{4}{3} \times 3.14 \times (5)^3 \]
First, calculate \( (5)^3 \):
\[ (5)^3 = 125 \]
Substitute it into the volume formula:
\[ V' = \frac{4}{3} \times 3.14 \times 125 \]
Calculate \( \frac{4}{3} \times 3.14 \):
\[ \frac{4}{3} \times 3.14 \approx 4.1888 \]
Now multiply by 125:
\[ V' \approx 4.1888 \times 125 \approx 523.6 \text{ in}^3 \]
-
Round the answer to the nearest tenth:
Rounding \( 523.6 \) gives \( 523.3 \).
Thus, the volume of the globe after reducing its dimensions by half is:
\[ \boxed{523.3 \text{ in}^3} \]
0 answers