The formula for the volume of a right circular cylinder is V = πr²h. If r = 2b and h = 5b + 3, what is the volume of the cylinder in terms of b?
Understand the Problem
The question is asking to calculate the volume of a cylinder given its radius and height in terms of variable b, utilizing the formula for the volume of a cylinder. The variables for radius and height are defined in terms of b, and we need to express the final volume also in terms of b.
Answer
$$ V = 20\pi b^3 + 12\pi b^2 $$
Answer for screen readers
The volume of the cylinder in terms of $b$ is
$$ V = 20\pi b^3 + 12\pi b^2 $$
Steps to Solve
- Identify the formula for volume The formula for the volume of a right circular cylinder is given by:
$$ V = \pi r^2 h $$
- Substitute the variables Given that the radius $r = 2b$ and the height $h = 5b + 3$, we can substitute these expressions into the volume formula:
$$ V = \pi (2b)^2 (5b + 3) $$
- Calculate $r^2$ We need to calculate $r^2$:
$$ (2b)^2 = 4b^2 $$
- Substitute $r^2$ into the volume formula Now we substitute $4b^2$ back into the volume formula:
$$ V = \pi \cdot 4b^2 \cdot (5b + 3) $$
- Distribute the expression Now, distribute $4b^2$ over $(5b + 3)$:
$$ V = \pi \cdot (4b^2 \cdot 5b + 4b^2 \cdot 3) $$
This results in:
$$ V = \pi (20b^3 + 12b^2) $$
- Write the final expression Thus, the final expression for volume in terms of $b$ is:
$$ V = 20\pi b^3 + 12\pi b^2 $$
The volume of the cylinder in terms of $b$ is
$$ V = 20\pi b^3 + 12\pi b^2 $$
More Information
The volume of a cylinder varies depending on the radius and height. By substituting values directly into the volume formula, we can express it in terms of another variable, which is useful for understanding relationships among dimensions.
Tips
- Forgetting to square the radius when plugging into the volume formula.
- Incorrectly distributing the terms when expanding the expression.
- Not substituting back correctly after simplification.
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