Answer

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**Hint:**The formula for the area of the circle is $\pi \left( {pie} \right)$ times the square of the radius. Use this to find the total area of both circles with radii $24cm$ and $7cm$. Now substitute the area in the formula and find the radius of the required circle. Then twice the radius is the diameter.

**Complete step-by-step answer:**Here in this problem, we are given a radii of two circles, i.e. $24cm$ and $7cm$ and with this data, we need to find the diameter of a circle which is having an area equal to the sum of the areas of circles with given radii.

So in order to do that, we first need to find out the total area of the circles having radii $24cm$ and $7cm$. Then with that value of the area, we can find the required value of diameter.

As we know, the area of a circle is the amount of two-dimensional space cover within the circumference of that circle. It can calculate by the square of radius time the value of $\pi \left( {pie} \right)$ .

$ \Rightarrow $ Area of a circle $ = \pi \times {\left( {{\text{Radius}}} \right)^2}$

Therefore, the area of two circles with radii $24cm$ and $7cm$ will be:

$ \Rightarrow $ The total area of the circles $ = \left( {\pi \times {{24}^2}} \right) + \left( {\pi \times {7^2}} \right)$

We also know the values ${24^2} = {\left( {12 \times 2} \right)^2} = {12^2} \times {2^2} = 144 \times 4 = 576$ and ${7^2} = 49$ . Using the squares in the above equation, we get:

$ \Rightarrow $ The total area of the circles $ = \left( {\pi \times {{24}^2}} \right) + \left( {\pi \times {7^2}} \right) = \pi \times \left( {{{24}^2} + {7^2}} \right) = \pi \times \left( {576 + 49} \right) = \left( {\pi \times 625} \right){\text{ c}}{{\text{m}}^2}$

Therefore, we get the area of the required circle as $\left( {\pi \times 625} \right){\text{ c}}{{\text{m}}^2}$

Now we need to find the radius of this circle with an area of $\left( {\pi \times 625} \right){\text{ c}}{{\text{m}}^2}$.

For this, we can again use the formula for the area of the circle, i.e. Area of a circle $ = \pi \times {\left( {{\text{Radius}}} \right)^2}$

$ \Rightarrow {\text{ Area }} = \pi \times Radiu{s^2} \Rightarrow \pi \times 625 = \pi \times {\left( {Radius} \right)^2}$

Now this equation can be further solved to find the unknown value of radius. After dividing both sides with $\pi \left( {pie} \right)$ :

$ \Rightarrow \pi \times 625 = \pi \times {\left( {Radius} \right)^2} \Rightarrow {\left( {Radius} \right)^2} = 625 \Rightarrow Radius = \sqrt {625} {\text{ }}cm$

Now to find the radius, we need to find the square root of the number $625$ .

We know that, $625 = 5 \times 5 \times 5 \times 5 = {\left( {5 \times 5} \right)^2} = {25^2}$

Therefore, we get:

$ \Rightarrow Radius = \sqrt {625} {\text{ }}cm = 25{\text{ }}cm$

The diameter of the required circle will be twice the radius, i.e. $Diameter{\text{ }} = 2 \times 25 = 50{\text{ }}cm$

**Note:**In questions like this the use of the correct formula to find the required quantity is necessary. Be careful while finding squares and square-roots of numbers. Notice in the solution, we calculated the total area of two circles as $\left( {\pi \times 625} \right){\text{ c}}{{\text{m}}^2}$ , without substituting the value of $\pi \left( {pie} \right)$ . This made the calculations less complicated since this $\pi \left( {pie} \right)$ got eliminated in the next step. Also, the use of distributive property in multiplication makes the calculations easy, i.e. $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ .

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