Question

The diameter of two circular plates are ${\text{10cm}}$ and ${\text{24cm}}$respectively. What is the diameter of the plate which has area equal to the combined area of the two given plates?

Answer 1

Hint: We can find the radius of two plates separately and find their area. The sum of their area will be equal to the area of the bigger plate. So, we can find the radius from the area and using the radius, we can find the diameter of the plate.

Complete step by step answer:

We have 2 circular plates and its diameters are given. We know that radius is half of the diameter. We can find
For plate with diameter ${\text{10cm}}$, we can find radius as,
${r_1} = \dfrac{d}{2} = \dfrac{{10}}{2} = 5cm$
We can calculate its area as,
${A_1} = \pi {r_1}^2 = \pi \times {5^2} = \pi \times 25c{m^2}$
For plate with diameter ${\text{24cm}}$, radius is given by,
${r_2} = \dfrac{d}{2} = \dfrac{{24}}{2} = 12cm$
We can calculate its area by,
\[{A_2} = \pi {r_2}^2 = \pi \times {12^2} = \pi \times 144c{m^2}\]
We can find the total area by adding the area of the two plates,
\[ \Rightarrow {\text{A = }}{{\text{A}}_{\text{1}}}{\text{ + }}{{\text{A}}_{\text{2}}}{\text{ = $\pi$ }}\left( {{\text{25 + 144}}} \right){\text{ = $\pi \times$ 169}}\]
We know that the combined area of the 2 plates is equal to the area of the bigger plate. Let r be the radius of the bigger plate. So, its area is given by,
\[
  A = \pi \times 169 \\
   \Rightarrow \pi {r^2} = \pi \times 169 \\
 \]
On dividing throughout with $\pi $, we get,
${r^2} = 169$
Taking the root, we get,
$r = \pm \sqrt {169} = \pm 13$
As the radius cannot be negative, we only take the positive square root.
$\therefore r = 13cm$
Therefore, diameter is given by, $d = 2r = 2 \times 13 = 26cm$
So, the diameter of the plate having the combined area is ${\text{26cm}}$.

Note: We only use the equations to find the area of a plate and diameter from its radius. We don’t need to find the actual value of the area as the term \[{\text{$\pi$ }}\] will get cancelled in the next equation. As \[{\text{$\pi$ }}\] is an irrational number, we need to take necessary approximations for calculating the actual area. Proper units must be used and units must be converted if necessary.