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The volume of a cylinder is given by the formula $V=\pi r^2 h$, where $r$ is the radius of the cylinder and $h$ is the height. Suppose a cylindrical can has radius $(x+8)$ and height $(2 x+3)$. Which expression represents the volume of the can?


A$\pi x^3+19 \pi x^2+112 \pi x+192 \pi$
B$2 \pi x^3+35 \pi x^2+80 \pi x+48 \pi$
C$2 \pi x^3+35 \pi x^2+176 \pi x+192 \pi$
D$4 \pi x^3+44 \pi x^2+105 \pi x+72 \pi$

Verified Answer :

  • Substitute the expressions for radius \(r = x + 8\) and height \(h = 2x + 3\) into the formula \(V = \pi r^2 h\).
  • Expand \((x + 8)^2\) to obtain \(x^2 + 16x + 64\).
  • Multiply the result by \((2x + 3)\) to get \(2x^3 + 35x^2 + 176x + 192\).
  • Multiply by \(\pi\) to reach the final volume: \(\boxed{2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi}\).

Explanation

  1. Problem Analysis and Substitution

The objective is to find the volume of a cylindrical can given its radius and height in terms of a variable \(x\). The volume \(V\) of a cylinder is calculated using the formula \(V = \pi r^2 h\). We are given:

  • The radius: \(r = x + 8\)
  • The height: \(h = 2x + 3\)

By substituting these expressions into the volume formula, we get \(V = \pi (x + 8)^2 (2x + 3)\).

  1. Expansion and Simplification

First, we square the expression for the radius:
\[(x + 8)^2 = x^2 + 2(x)(8) + 8^2 = x^2 + 16x + 64.\]
Now we multiply this result by the height \((2x + 3)\):
\[(x^2 + 16x + 64)(2x + 3) = x^2(2x + 3) + 16x(2x + 3) + 64(2x + 3).\]
Distributing the terms:
\[= (2x^3 + 3x^2) + (32x^2 + 48x) + (128x + 192).\]
Combining the like terms:
\[= 2x^3 + (3x^2 + 32x^2) + (48x + 128x) + 192 = 2x^3 + 35x^2 + 176x + 192.\]

  1. Final Volume Calculation

Finally, we multiply the resulting polynomial by \(\pi\) to get the volume \(V\):
\[V = \pi (2x^3 + 35x^2 + 176x + 192) = 2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi.\]
Comparing this with the given options, we find it matches the third expression. Therefore, the volume of the can is \(\boxed{2 \pi x^3 + 35 \pi x^2 + 176 \pi x + 192 \pi}\).

Examples

In engineering and manufacturing, understanding how volume changes as a function of dimensions is crucial for material efficiency. For example, if a company is designing a beverage can, they need to optimize the amount of metal used for the surface area while maintaining a specific volume capacity. By representing these dimensions as algebraic expressions in terms of a variable, they can model how design adjustments to the radius or height impact the overall volume and, subsequently, the manufacturing costs.

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