To find the volumes of the original cone and the dilated cone, we need to use the formula for the volume of a cone, which is:
\[ V = \frac{1}{3} \pi r^2 h \]
Step 1: Calculate the original volume of the cone
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The diameter of the base is 8 feet, so the radius \( r \) is half of that: \[ r = \frac{8}{2} = 4 \text{ feet} \]
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The height \( h \) is given as 9 feet.
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Plugging in the values into the volume formula: \[ V = \frac{1}{3} \pi (4)^2 (9) = \frac{1}{3} \pi (16)(9) = \frac{1}{3} \pi (144) = 48 \pi \text{ cubic feet} \]
Step 2: Calculate the volume of the dilated cone
When the cone is dilated using a scale factor of \( \frac{1}{4} \), the new dimensions will be:
- New radius: \[ r' = \frac{1}{4} \times 4 = 1 \text{ foot} \]
- New height: \[ h' = \frac{1}{4} \times 9 = \frac{9}{4} \text{ feet} \]
Now, we calculate the volume of the dilated cone:
- Using the volume formula again: \[ V' = \frac{1}{3} \pi (1)^2 \left(\frac{9}{4}\right) = \frac{1}{3} \pi (1) \left(\frac{9}{4}\right) = \frac{1}{3} \times \frac{9}{4} \pi = \frac{3}{4} \pi \text{ cubic feet} \]
Step 3: Compare the volumes
- Original volume of the cone: \( 48 \pi \)
- Dilated volume of the cone: \( \frac{3}{4} \pi \)
Conclusion
The correct response is:
The original volume of the cone is \( 48 \pi \) and the dilated volume is much smaller, \( \frac{3}{4} \pi \).
Thus, the answer is:
"The original volume of the cone is 48π and the dilated volume is much smaller, \( \frac{3}{4} \pi \)."
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