Question
3. Complete the final part of the job application by writing an expression for the volume of a cylinder with radius \( 3 x^{2} y \) and height \( 2 x y \). Use the formula \( \mathrm{V}=\pi r^{2} h \) where \( r \) is the radius and \( h \) is the height. Simplify your answer as much as possible.
Ask by Schofield Patel. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The volume of the cylinder is \( 18\pi x^{5} y^{3} \).
Solution
The volume of a cylinder is given by V = πr²h. For a cylinder with radius r = 3x²y and height h = 2xy, substitute these values into the formula:
V = π(3x²y)²(2xy).
First, square the radius:
(3x²y)² = 9x⁴y².
Now, multiply by the height:
V = π · 9x⁴y² · 2xy = 18π x⁵y³.
Thus, the simplified expression for the volume is 18πx⁵y³.
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Simplify this solution The Deep Dive
To find the volume \( V \) of the cylinder, we use the formula \( V = \pi r^2 h \). Plugging in the values for the radius \( r = 3x^2y \) and height \( h = 2xy \): 1. Calculate \( r^2 \): \[ r^2 = (3x^2y)^2 = 9x^4y^2 \] 2. Now substitute \( r^2 \) and \( h \) into the volume formula: \[ V = \pi (9x^4y^2)(2xy) \] 3. Multiply the terms: \[ V = \pi (9 \times 2)x^4y^2 \times xy = \pi (18)x^4y^2 \times xy \] 4. Simplifying further, we have: \[ V = 18\pi x^{4+1} y^{2+1} = 18\pi x^5y^3 \] So, the final simplified expression for the volume of the cylinder is: \[ V = 18\pi x^5y^3 \]
