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The surface area of a sphere is the same as the curved surface area of a cone having the radius of the base as \[120cm\] and height \[160cm\]. Find the radius of the sphere.

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Last updated date: 13th Jun 2024
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Answer
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Hint: Write down the given and the formulas of the for the surface area of the sphere and the curved surface area of a cone. Substitute the values of the given measurements and equate them, and solve the linear equation to get the value of the asked variable.

Complete step-by-step solution:
Curved surface area of a cone with radius $120cm$ and height $160cm$ is given as
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Let us consider the radius of the sphere to be \[r\].
Given,
Base of the cone \[ = 120cm\]
Base of the cone can also be taken as the radius of the cone because the base of the cone is a circle.
\[ \Rightarrow \]Radius of the cone \[ = R = 120cm\]
Height of the cone \[ = 160cm\]
The surface area of the sphere is the same as the curved surface area of a cone.
\[ \Rightarrow \]The surface area of the sphere \[ = \] the curved surface area of a cone.
The surface area of a sphere \[ = 4\pi {r^2}\]
The curved surface area of a cone \[ = \pi Rl\]
Here, \[l\] is the length of the cone. Because the length is unknown, we write it in terms of the given values, i.e., in terms of radius and height of the cone.
Since the cone is in the shape of the triangle observed in a two-dimension, we draw a perpendicular through the middle of the cone. When it becomes a right-angled triangle in a two-dimensional plane, we can get the length of the cone using Pythagoras theorem. We have,
\[l = \sqrt {{R^2} + {h^2}} \]
Substituting the value in the formula of the curved surface area of a cone, we get;
The curved surface area of a cone\[ = \pi R\sqrt {{R^2} + {h^2}} \]
Given,
\[4\pi {r^2} = \pi R\sqrt {{R^2} + {h^2}} \]
Substituting the values of the variables from the given question, we get;
\[4\pi {r^2} = \pi \times 120\sqrt {{{\left( {120} \right)}^2} + {{\left( {160} \right)}^2}} \]
Cancelling out the common terms, that is, pi, and simplifying the right-hand side, we get;
\[4{r^2} = 120\sqrt {14400 + 25600} \]
Simplifying the equation, we get;
\[{r^2} = 30\sqrt {40000} \]
Taking the perfect square out, we get;
\[ \Rightarrow {r^2} = 30 \times 200\]
Multiplying the right-hand side, we get;
\[ \Rightarrow {r^2} = 6000\]
Applying square root on both the sides, we get;
\[r = \sqrt {6000} \]
Simplifying the square root, we get;
\[r = 77.45\]

The radius of the sphere is 77.45.

Note: A sphere is defined as the geometrical object which is only in a three- dimensional space. It resembles that of a ball and when viewed in two-dimensional resembles a circle. A cone as well, is a three- dimensional shape in geometry that tapers smoothly from a flat base of a circle into a cone.