Cylinder Volume Percentage Change Calculation
We need to find the percentage increase in the volume of a right circular cylinder when its radius decreases by 19% and its height increases by 129%.
Volume Formula and Initial Values
The volume (\(V\)) of a right circular cylinder is given by the formula:
\( V = \pi r^2 h \)
Let the initial radius be \(r_1\) and the initial height be \(h_1\). The initial volume is:
\( V_1 = \pi r_1^2 h_1 \)
Calculating New Dimensions
The radius is decreased by 19%. The new radius (\(r_2\)) is:
\( r_2 = r_1 - (0.19 \times r_1) = r_1 (1 - 0.19) = 0.81 r_1 \)
The height is increased by 129%. The new height (\(h_2\)) is:
\( h_2 = h_1 + (1.29 \times h_1) = h_1 (1 + 1.29) = 2.29 h_1 \)
Calculating New Volume
The new volume (\(V_2\)) using the new dimensions is:
\( V_2 = \pi r_2^2 h_2 \)
Substitute the expressions for \(r_2\) and \(h_2\):
\( V_2 = \pi (0.81 r_1)^2 (2.29 h_1) \)
\( V_2 = \pi (0.81^2 r_1^2) (2.29 h_1) \)
\( V_2 = \pi (0.6561 r_1^2) (2.29 h_1) \)
\( V_2 = (0.6561 \times 2.29) (\pi r_1^2 h_1) \)
Calculate the product \(0.6561 \times 2.29\):
\( 0.6561 \times 2.29 \approx 1.502469 \)
So, the new volume is approximately:
\( V_2 \approx 1.502469 V_1 \)
Determining Percentage Increase
The percentage increase in volume is calculated as:
\( \text{Percentage Increase} = \frac{V_2 - V_1}{V_1} \times 100 \% \)
\( \text{Percentage Increase} = \frac{1.502469 V_1 - V_1}{V_1} \times 100 \% \)
\( \text{Percentage Increase} = \frac{(1.502469 - 1) V_1}{V_1} \times 100 \% \)
\( \text{Percentage Increase} = 0.502469 \times 100 \% \)
\( \text{Percentage Increase} \approx 50.2469 \% \)
Rounding to the nearest integer value, the percentage increase is 50%.
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