The tool maker was asked to open the round gate from 0.030 inch to 0.040 inch. What was the percent change of the area of the gate?
Understand the Problem
The question asks to calculate the percentage change in the area of a round gate when its radius changes from 0.030 inches to 0.040 inches. Since the area of a circle is proportional to the square of its radius, we need to calculate the areas with both radii, and then find the percentage change between the two areas.
Answer
77.79%
Answer for screen readers
77.79%
Steps to Solve
- Calculate the initial area
The formula for the area of a circle is $A = \pi r^2$. The initial radius is 0.030 inches. So, the initial area $A_1$ is: $A_1 = \pi (0.030)^2 = \pi (0.0009) = 0.0009\pi$
- Calculate the final area
The final radius is 0.040 inches. So, the final area $A_2$ is: $A_2 = \pi (0.040)^2 = \pi (0.0016) = 0.0016\pi$
- Calculate the change in area
The change in area is the difference between the final and initial areas: $\Delta A = A_2 - A_1 = 0.0016\pi - 0.0009\pi = 0.0007\pi$
- Calculate the percentage change in area
The percentage change is calculated as: Percentage Change $= \frac{\text{Change in Area}}{\text{Initial Area}} \times 100$ Percentage Change $= \frac{0.0007\pi}{0.0009\pi} \times 100 = \frac{0.0007}{0.0009} \times 100 = \frac{7}{9} \times 100 \approx 77.777...$
- Rounding to two decimal places Rounding the percentage change to two decimal places, we get 77.78%.
77.79%
More Information
The percent change of the area can be calculated directly by just squaring the radii $ (\frac{0.040^2 - 0.030^2}{0.030^2}) \times 100 $ since $\pi$ is present in both the numerator and denominator.
Tips
A common mistake is to calculate the percentage change in the radius and then assume that the percentage change in the area is the same. This is incorrect because the area is proportional to the square of the radius. Another mistake is to not account for the initial area when performing the percent change calculation $ (\frac{\text{new - old}}{\text{old}})$, which is crucial for calculating the percentage change.
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