Understanding Cone Volume Changes
This problem asks us to find the percentage change in the volume of a right circular cone when its dimensions, specifically the radius and height, are altered. We need to calculate how the volume changes when the radius increases by 20% and the height decreases by 25%.
Cone Volume Formula
The formula for the volume ($V$) of a right circular cone is:
$$ V = \frac{1}{3} \pi r^2 h $$
where '$r$' is the radius of the base and '$h$' is the height of the cone.
Calculating New Dimensions
Let the original radius be '$r_1$' and the original height be '$h_1$'. The original volume is:
$$ V_1 = \frac{1}{3} \pi r_1^2 h_1 $$
Now, let's calculate the new dimensions:
- Radius Change: The radius is increased by 20%. The new radius, '$r_2$', is: $$ r_2 = r_1 + 0.20 r_1 = 1.20 r_1 $$
- Height Change: The height is decreased by 25%. The new height, '$h_2$', is: $$ h_2 = h_1 - 0.25 h_1 = 0.75 h_1 $$
Calculating New Volume
The new volume, '$V_2$', using the new radius '$r_2$' and new height '$h_2$', is:
$$ V_2 = \frac{1}{3} \pi r_2^2 h_2 $$
Substitute the expressions for '$r_2$' and '$h_2$' into the formula:
$$ V_2 = \frac{1}{3} \pi (1.20 r_1)^2 (0.75 h_1) $$
Simplify the expression:
$$ V_2 = \frac{1}{3} \pi (1.44 r_1^2) (0.75 h_1) $$
Rearrange the terms to compare with the original volume '$V_1$':
$$ V_2 = \left( \frac{1}{3} \pi r_1^2 h_1 \right) \times (1.44 \times 0.75) $$
$$ V_2 = V_1 \times (1.44 \times 0.75) $$
Determining Volume Change Factor
Let's calculate the product of the factors for the radius squared and height:
$$ 1.44 \times 0.75 = 1.08 $$
So, the new volume is:
$$ V_2 = V_1 \times 1.08 $$
This means the new volume is 1.08 times the original volume.
Calculating Percentage Increase in Volume
To find the percentage increase, we calculate the difference between the new and original volumes and divide by the original volume, then multiply by 100%:
Percentage Increase $= \frac{V_2 - V_1}{V_1} \times 100\%$
Substitute '$V_2 = 1.08 V_1$':
Percentage Increase $= \frac{1.08 V_1 - V_1}{V_1} \times 100\%$
Percentage Increase $= \frac{0.08 V_1}{V_1} \times 100\%$
Percentage Increase $= 0.08 \times 100\%$
Percentage Increase $= 8\% $
Conclusion
Therefore, when the radius of the right circular cone is increased by 20% and its height is decreased by 25%, the volume of the cone increases by 8%.
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