To find the percentage reduction in the volume of the container when the dimensions are each reduced by half an inch, we will follow these steps:
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Calculate the Original Volume: The container is a cylinder, with the volume \( V \) calculated using the formula: \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height.
Given:
- Height \( h = 8 \) inches
- Diameter \( d = 4 \) inches, thus the radius \( r = \frac{d}{2} = \frac{4}{2} = 2 \) inches.
Now, substituting the values into the volume formula: \[ V_{\text{original}} = \pi (2)^2 (8) = \pi (4)(8) = 32\pi , \text{cubic inches}. \]
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Calculate the New Volume: When both dimensions are reduced by half an inch:
- New height \( h_{\text{new}} = 8 - 0.5 = 7.5 \) inches
- New radius \( r_{\text{new}} = 2 - 0.5 = 1.5 \) inches.
Now calculate the new volume: \[ V_{\text{new}} = \pi (1.5)^2 (7.5) = \pi (2.25)(7.5) = 16.875\pi , \text{cubic inches}. \]
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Calculate the Volume Reduction: The volume reduction is: \[ \Delta V = V_{\text{original}} - V_{\text{new}} = 32\pi - 16.875\pi = (32 - 16.875)\pi = 15.125\pi , \text{cubic inches}. \]
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Calculate the Percentage Reduction: The percentage reduction in volume can be calculated as: \[ \text{Percentage Reduction} = \left( \frac{\Delta V}{V_{\text{original}}} \right) \times 100% = \left( \frac{15.125\pi}{32\pi} \right) \times 100% = \left( \frac{15.125}{32} \right) \times 100%. \]
\[ \text{Percentage Reduction} \approx \left( 0.47265625 \right) \times 100% \approx 47.27% \approx 47% \text{ (rounded to the nearest percentage)}. \]
Thus, the percentage reduction in the volume of the container is approximately 47%.
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