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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,