Answer
Verified
435.9k+ views
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. $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ .
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. $a \times \left( {b + c} \right) = \left( {a \times b} \right) + \left( {a \times c} \right)$ .
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
One Metric ton is equal to kg A 10000 B 1000 C 100 class 11 physics CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE
Give 10 examples for herbs , shrubs , climbers , creepers