Solve the following formula for $r$
$V=\frac{\pi r^2 h}{3}$
A. $r=\sqrt{\frac{h \pi}{3 V}}$
B. $r=\sqrt{\frac{6 V}{h \pi}}$
C. $r=\sqrt{\frac{3 V}{h^2 \pi}}$
D. $r=\sqrt{\frac{3 V}{h \pi}}$
Answer (2)
Multiply both sides of the equation by 3: 3 V = π r 2 h .
Divide both sides by πh : πh 3 V = r 2 .
Take the square root of both sides: r = πh 3 V .
The solution for r is: r = πh 3 V .
Explanation
Understanding the Problem We are given the formula V = 3 π r 2 h and we need to solve for r . This means we want to isolate r on one side of the equation.
Multiply by 3 First, let's multiply both sides of the equation by 3 to get rid of the fraction: 3 V = π r 2 h
Divide by πh Next, we want to isolate r 2 , so we divide both sides of the equation by πh :
πh 3 V = r 2
Take the square root Finally, to solve for r , we take the square root of both sides of the equation: r = πh 3 V
Examples
Imagine you are designing a conical paper cup for a water dispenser. You know the desired volume V of the cup and the height h . By solving the formula for the volume of a cone for the radius r , you can determine the radius needed to achieve the desired volume with the given height. This ensures that the paper cup will hold the right amount of water, preventing spills and waste. Knowing how to manipulate formulas allows for precise design and efficient use of materials in various engineering and design applications.
To solve for r in the volume formula of a cone, we rearranged the equation to isolate r and found r = πh 3 V . This corresponds to option D. The steps involved multiplying by 3, dividing by πh , and taking the square root.
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