Question

# Explain it clearly and easy to understand. Show step-by-step. (In this

Circles
Explain it clearly and easy to understand. Show step-by-step. (In this in case, $$\displaystyle\pi={3.1416}$$) Round the answer to the nearest tenth.
There are two circles. The diameter of the first is 12 in. and the area of the second is $$\displaystyle{2}{\frac{{{1}}}{{{4}}}}$$ times the area of the first. What is the diameter of the second circle?

2021-08-10

Step 1
Let d and $$\displaystyle{A}_{{{1}}}$$ be the diameter and area of first circle and D and $$\displaystyle{A}_{{{2}}}$$ be diameter and area of second circle.
Then , according to given condition,
$$\displaystyle{d}={12}$$
and
$$\displaystyle{A}_{{{2}}}={2}{\frac{{{1}}}{{{4}}}}{A}_{{{1}}}$$
$$\displaystyle{A}_{{{2}}}={\frac{{{9}}}{{{4}}}}{A}_{{{1}}}$$
We know that area of circle is $$\pi r^2$$ , that is $$\pi \frac{dr^2}{2}$$
Step 2
Therefore, $$\displaystyle{A}_{{{2}}}={\frac{{{9}}}{{{4}}}}{A}_{{{1}}}$$
$$\displaystyle\pi{\left({\frac{{{D}}}{{{2}}}}\right)}^{{{2}}}={\frac{{{9}}}{{{4}}}}\pi{\left({\frac{{{d}}}{{{2}}}}\right)}^{{{2}}}$$
$$\displaystyle{D}^{{{2}}}={\frac{{{9}}}{{{4}}}}{d}^{{{2}}}$$
$$\displaystyle{D}={\frac{{{3}}}{{{2}}}}{d}$$
$$\displaystyle{D}={\frac{{{3}}}{{{2}}}}{\left({12}\right)}={24}$$ in.
Step 3
Answer: Diameter of second circle is 24 in.