Question

A sphere has a diameter of 4,320 meters. How many meters long is "Unit Y" if the surface of the sphere, measured in square Units Y is equal to the volume of the sphere measured in cube Units Y.

Answer 1

Task:

A sphere has a diameter of 4.320 meters. How many meters long is "Unit Y" if the surface of the sphere, measured in square Units Y is equal to the volume of the sphere measured in cube Units Y.

Solution:

Sphere

Surface Area

Volume

$V = \frac{4}{3}\pi r^3$

$2r = 4.32m$

$4\pi r^{2} = 58.630m^{2} = X(UnitY)^{2}$

$\frac{4}{3}\pi r^3 = 42.213m^3 = X(UnitY)^3$

Unit $Y = \left(\frac{58.630m^2}{X}\right)^{\frac{1}{2}}$

Unit $Y = \left(\frac{42.213m^3}{X}\right)^{\frac{1}{3}}$

$\left(\frac{58.630m^2}{X}\right)^{\frac{1}{2}} = \left(\frac{42.213m^3}{X}\right)^{\frac{1}{3}}$

$\frac{(58.630)^{\frac{1}{2}}}{(X)^{\frac{1}{2}}} = \frac{(42.213)^{\frac{1}{3}}}{(X)^{\frac{1}{3}}}$

$(42.213)^{\frac{1}{3}}\cdot (X)^{\frac{1}{2}} = (58.630)^{\frac{1}{2}}\cdot (X)^{\frac{1}{3}}$

$(X)^{\frac{1}{6}} = \frac{(58.630)^{\frac{1}{2}}}{(42.213)^{\frac{1}{3}}}$

$X = \left(\frac{(58.630)^{\frac{1}{2}}}{(42.213)^{\frac{1}{3}}}\right)^{6} = \frac{58.630^{3}}{42.213^{2}} = 113.101$

Answer:

$X = 113.101$