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V=(4)/(3)pir^(3)
The volume,
V, of a sphere of radius,
r, can be found with the given equation. Which of the following correctly shows the sphere's radius in terms of its volume?
Choose 1 answer:
(A)
r=(3)/(4)piV^((1)/(3))
(B)
r=(4)/(3)piV^((1)/(3))
(c)
r=((3V)/(4pi))^((1)/(3))
(D)
r=((4V)/(3pi))^((1)/(3))

$V=\frac{4}{3} \pi r^{3}$ $\newline$ The volume, $V$ , of a sphere of radius, $r$ , can be found with the given equation. Which of the following correctly shows the sphere's radius in terms of its volume? $\newline$ Choose $1$ answer: $\newline$ (A) $r=\frac{3}{4} \pi V^{\frac{1}{3}}$ $\newline$ (B) $r=\frac{4}{3} \pi V^{\frac{1}{3}}$ $\newline$ (c) $r=\left(\frac{3 V}{4 \pi}\right)^{\frac{1}{3}}$ $\newline$ (D) $r=\left(\frac{4 V}{3 \pi}\right)^{\frac{1}{3}}$

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Compare linear and exponential growth

Full solution

Q.

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

\newline

The volume,

V

, of a sphere of radius,

r

, can be found with the given equation. Which of the following correctly shows the sphere's radius in terms of its volume?

\newline

Choose

1

answer:

\newline

(A)

r=\frac{3}{4} \pi V^{\frac{1}{3}}

\newline

(B)

r=\frac{4}{3} \pi V^{\frac{1}{3}}

\newline

(c)

r=\left(\frac{3 V}{4 \pi}\right)^{\frac{1}{3}}

\newline

(D)

r=\left(\frac{4 V}{3 \pi}\right)^{\frac{1}{3}}

Write volume formula for sphere: Write down the given volume formula for a sphere. $\newline$ The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$ .
Solve formula for radius: Solve the formula for $r$ (the radius). $\newline$ To find $r$ in terms of $V$ , we need to isolate $r$ on one side of the equation. We start by multiplying both sides by $\frac{3}{4}$ to cancel out the $\frac{4}{3}$ on the right side. $\newline$ $\frac{3}{4}V = \pi r^3$
Divide both sides by $\pi$ : Divide both sides by $\pi$ to isolate $r^3$ . $\newline$ $\left(\frac{3}{4}V\right)/\pi = r^3$
Take cube root to solve for $r$ : Take the cube root of both sides to solve for $r$ . $\newline$ $r = \left(\frac{\left(\frac{$ $3$ }{ $4$ }\right)V}{\pi}\right)^{\frac{ $1$ }{ $3$ }}
Simplify expression for r: Simplify the expression for r. $\newline$ $r = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}$
Match expression with choices: Match the simplified expression for $r$ with the given choices. $\newline$ The correct expression that matches our result is (C) $r = \left(\frac{3V}{4\pi}\right)^{\frac{1}{3}}$ .

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