To find the approximate difference in volumes of a youth softball and an adult softball, we will use the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius of the sphere. The radius is half of the diameter.
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Calculate the radius of the youth softball: \[ r_{\text{youth}} = \frac{3.5}{2} = 1.75 \text{ in} \]
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Calculate the volume of the youth softball: \[ V_{\text{youth}} = \frac{4}{3} \pi (1.75)^3 \] Calculating \( (1.75)^3 \): \[ (1.75)^3 = 5.359375 \]
Now, substitute into the volume formula: \[ V_{\text{youth}} = \frac{4}{3} \times 3.14 \times 5.359375 \] \[ V_{\text{youth}} \approx \frac{4 \times 3.14 \times 5.359375}{3} \] \[ V_{\text{youth}} \approx \frac{67.1908225}{3} \approx 22.39694083 \text{ in}^3 \]
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Calculate the radius of the adult softball: \[ r_{\text{adult}} = \frac{3.8}{2} = 1.9 \text{ in} \]
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Calculate the volume of the adult softball: \[ V_{\text{adult}} = \frac{4}{3} \pi (1.9)^3 \] Calculating \( (1.9)^3 \): \[ (1.9)^3 = 6.859 \]
Now, substitute into the volume formula: \[ V_{\text{adult}} = \frac{4}{3} \times 3.14 \times 6.859 \] \[ V_{\text{adult}} \approx \frac{4 \times 3.14 \times 6.859}{3} \] \[ V_{\text{adult}} \approx \frac{86.21352}{3} \approx 28.73784 \text{ in}^3 \]
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Calculate the difference in volumes: \[ \text{Difference} = V_{\text{adult}} - V_{\text{youth}} \] \[ \text{Difference} \approx 28.73784 - 22.39694083 \approx 6.34089917 \text{ in}^3 \]
Rounding this to the nearest tenth gives: \[ \text{Difference} \approx 6.3 \text{ in}^3 \]
Thus, the approximate difference in their volumes is 6.3 in³.