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Chapter 1: Problem 10

If the radius of a circular garden plot is increased by \(25 \%\) by what percentage does the area of the garden increase?

Short Answer

Expert verified
Answer: The area of the circular garden increases by approximately 56.25% when the radius is increased by 25%.

Step by step solution

01

Calculate the original area

Let the original radius of the circular garden plot be \(r\). We can label the original area as \(A_1\). The area of the circle is given by the formula \(A = \pi r^2\). So we can write the original area as \(A_1 = \pi r^2\).
02

Calculate the new radius

We are given that the radius is increased by 25%. We can express this increase as a multiplication factor: \(1 + 0.25 = 1.25\). Therefore, the new radius, \(r'\), can be calculated as follows: \(r' = 1.25r\).
03

Calculate the new area

Now that we have the new radius, let's find the new area, \(A_2\). Using the formula for the area of a circle with the new radius, we get \(A_2 = \pi (r')^2 = \pi (1.25r)^2\).
04

Calculate the area increase

Next, we want to find the increase in the area. We can do this by subtracting the original area \(A_1\) from the new area \(A_2\). This gives us: \(\Delta A = A_2 - A_1 = \pi (1.25r)^2 - \pi r^2\).
05

Calculate the percentage increase

Finally, we can find the percentage increase in the area by dividing the area increase, \(\Delta A\), by the original area, \(A_1\), and then multiplying by 100%. This can be written as: \(\text{Percentage Increase} = \frac{\Delta A}{A_1} \times 100\%\). Substitute the expressions for \(\Delta A\) and \(A_1\) and simplify: \(\text{Percentage Increase} = \frac{\pi (1.25r)^2 - \pi r^2}{\pi r^2} \times 100\% = \frac{(1.25^2 - 1)r^2}{r^2} \times 100\% = (1.25^2 - 1) \times 100\% \approx 56.25\%\) The area of the circular garden increases by approximately 56.25% when the radius is increased by 25%.

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