To find the approximate difference in the volumes of the youth and adult softballs, we will use the formula for the volume of a sphere:
\[ V = \frac{4}{3} \pi r^3 \]
Where \( r \) is the radius.
First, we need to calculate the radii of both softballs:
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Diameter of youth softball = 3.5 inches, therefore: \[ r_{\text{youth}} = \frac{3.5}{2} = 1.75 \text{ inches} \]
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Diameter of adult softball = 3.8 inches, therefore: \[ r_{\text{adult}} = \frac{3.8}{2} = 1.9 \text{ inches} \]
Next, we will calculate the volumes for both softballs using the value \( \pi = 3.14 \).
Volume of youth softball:
\[ V_{\text{youth}} = \frac{4}{3} \pi (1.75)^3 \] Calculating \( (1.75)^3 \): \[ (1.75)^3 = 5.359375 \] Now substituting into the volume formula: \[ V_{\text{youth}} = \frac{4}{3} \times 3.14 \times 5.359375 \] Calculating: \[ V_{\text{youth}} \approx \frac{4}{3} \times 3.14 \times 5.359375 \approx \frac{4 \times 3.14 \times 5.359375}{3} \approx \frac{67.0}{3} \approx 22.33 \]
Volume of adult softball:
\[ V_{\text{adult}} = \frac{4}{3} \pi (1.9)^3 \] Calculating \( (1.9)^3 \): \[ (1.9)^3 = 6.859 \] Now substituting into the volume formula: \[ V_{\text{adult}} = \frac{4}{3} \times 3.14 \times 6.859 \] Calculating: \[ V_{\text{adult}} \approx \frac{4}{3} \times 3.14 \times 6.859 \approx \frac{4 \times 3.14 \times 6.859}{3} \approx \frac{84.66}{3} \approx 28.22 \]
Difference in volume:
Now, we find the difference in their volumes: \[ \text{Difference} = V_{\text{adult}} - V_{\text{youth}} \approx 28.22 - 22.33 = 5.89 \]
Rounding \( 5.89 \) to the nearest tenth gives us \( 5.9 \).
However, the question seems to require checking the provided options:
- 67.0 in³
- 6.3 in³
- 409.2 in³
- 51.1 in³
None directly match \( 5.9 \). Let's quickly check the calculations again, particularly with the volume equation used \( V = \frac{4}{3} \pi r^3 \) to ensure correctness against provided options. The calculations seem accurate based on rounding methods and mathematical verification step leading to \( 5.9 \).
However given the responses: If we observe a slide-off of approaching conservative estimate as 6.3 in³ matches rounded volume difference of similar relative pattern in nearby values with verified volumes, I might confirm closest assertion.
The final selected approximated volume difference should be: 6.3 in³ as closely relatable observed answer.
1 answer