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Find the radius of a sphere whose circumference and solid content have the same numerical value.

Last updated date: 13th Jul 2024
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Hint: First we have to define what the terms we need to solve the problem are.
Since we need to find the radius of the sphere, and things we need to know about circumference first which is the perimeter of the given circle or else like the ellipse. And the solid content is the total amount of the base product. sphere radius that we need to find using $V = \dfrac{{4\pi {r^3}}}{3}$(pie is twenty two divides seven)

Complete step-by-step solution:
Thus, we need to find the radius of the sphere for the given problem and let take or assume that $r$is the radius of the sphere, so that it will useful in the calculation purpose;
Now since the radius is half of the diameter. And the sphere radius is known as r which has to find using the $V = \dfrac{{4\pi {r^3}}}{3}$ and also that $2\pi r$ is the circumference of a given circle or perimeter with the radius $r$.
To find the same numerical value we just need to equals to above two known values and find the radius for it; which means $\dfrac{{4\pi {r^3}}}{3} = 2\pi r$(to find the radius of the sphere for same numerical value so we equivalent that both equations)
Hence solving further, we can able to obtain; $\dfrac{{4\pi {r^3}}}{3} = 2\pi r \Rightarrow \dfrac{{4{r^2}}}{3} = 2$(both sides pie and r will cancel each other) and hence $ \Rightarrow \dfrac{{4{r^2}}}{3} = 2 \Rightarrow {r^2} = \dfrac{{2 \times 3}}{4}$(cross multiplying we get)
Hence, we get ${r^2} = 1.5$(now taking square roots on the both we get in left side only r as the radius so we are doing that)
Thus ${r^2} = 1.5 \Rightarrow r = 1.22$which is the required radius of the sphere whose circumference and the solid content have the same numerical value as well as.

Note: since radius of the sphere is $V = \dfrac{{4\pi {r^3}}}{3}$and circumference of the given circle or perimeter is $2\pi r$
Also, radius means half of the given diameter in the circle or sphere; numerical means the radius will need to be shown as the number value.