Problem 78
Question
For the following exercises, write the polynomial function that models the given situation. A cylinder has a radius of \(x+2\) units and a height of 3 units greater. Express the volume of the cylinder as a polynomial function.
Step-by-Step Solution
Verified Answer
The volume of the cylinder as a polynomial is \( V(x) = \pi (x^3 + 9x^2 + 24x + 20) \).
1Step 1: Understand the Cylinder's Volume Formula
The volume \( V \) of a cylinder is given by the formula \( V = \pi \times r^2 \times h \) where \( r \) is the radius and \( h \) is the height of the cylinder.
2Step 2: Identify the Radius and Height
In this problem, the radius \( r \) is \( x+2 \) units and the height \( h \) is 3 units greater than the radius, so \( h = (x+2) + 3 = x+5 \) units.
3Step 3: Substitute into the Volume Formula
Substitute the expressions for \( r \) and \( h \) into the volume formula:\[V = \pi \times (x+2)^2 \times (x+5)\]
4Step 4: Expand the Radius Squared Term
First, expand \((x+2)^2\):\[(x+2)^2 = x^2 + 4x + 4\]
5Step 5: Expand the Whole Polynomial
Substitute \(x^2 + 4x + 4\) back into the volume equation and distribute across \(x+5\):\[V = \pi \times (x^2 + 4x + 4) \times (x+5)\]Expand this by distributing:\[V = \pi \times (x^3 + 5x^2 + 4x^2 + 20x + 4x + 20)\]Combine like terms:\[V = \pi \times (x^3 + 9x^2 + 24x + 20)\]
6Step 6: Write the Polynomial Function
Thus, the polynomial function for the volume of the cylinder is:\[ V(x) = \pi (x^3 + 9x^2 + 24x + 20) \]
Key Concepts
Cylinder VolumePolynomial ExpansionCylinder DimensionsAlgebraic Modeling
Cylinder Volume
The volume of a cylinder is crucial in understanding how much space is inside this 3D shape. To find the volume, use the formula:
Each of these components plays a role in determining the space inside the cylinder. In practice, finding the volume means calculating the area of the base (a circle in this case) and then multiplying it by the height of the cylinder.
In this exercise, we use algebraic expressions to represent the radius and height, which allows us to explore the concept in greater depth by incorporating polynomial expressions.
- For volume, use: \( V = \pi \times r^2 \times h \)
Each of these components plays a role in determining the space inside the cylinder. In practice, finding the volume means calculating the area of the base (a circle in this case) and then multiplying it by the height of the cylinder.
In this exercise, we use algebraic expressions to represent the radius and height, which allows us to explore the concept in greater depth by incorporating polynomial expressions.
Polynomial Expansion
Polynomial expansion can seem tricky at first, but it simply involves expressing a product of polynomials as a sum of simpler terms.
When expanding a polynomial like \((x+2)^2\), you'll need to apply the distributive property (often referred to as FOIL for binomials).
Here, you multiply each term in the first binomial by each term in the second:
Once expanded, you have a polynomial that is ready for further multiplication if needed.
Practicing these expansions is key to mastering polynomial manipulations, which are critical in many algebraic modeling situations.
When expanding a polynomial like \((x+2)^2\), you'll need to apply the distributive property (often referred to as FOIL for binomials).
Here, you multiply each term in the first binomial by each term in the second:
- \((x+2)(x+2) = x^2 + 2x + 2x + 4 = x^2 + 4x + 4\)
Once expanded, you have a polynomial that is ready for further multiplication if needed.
Practicing these expansions is key to mastering polynomial manipulations, which are critical in many algebraic modeling situations.
Cylinder Dimensions
The dimensions of a cylinder directly influence its volume. In this exercise, we're given the radius and height in terms of algebraic expressions.
Specifically, the radius is given as \(x+2\), and the height is described as being 3 units more than the radius, leading to \(x+5\).
Understanding how to derive additional dimensions from given ones is a fundamental skill in algebraic modeling.
Specifically, the radius is given as \(x+2\), and the height is described as being 3 units more than the radius, leading to \(x+5\).
Understanding how to derive additional dimensions from given ones is a fundamental skill in algebraic modeling.
- This might involve recognizing patterns like finding height as a function of radius or any other dimensions.
Algebraic Modeling
Algebraic modeling involves using algebra to represent real-world scenarios, turning them into mathematical expressions.
In this task, we translated the physical dimensions of a cylinder into an algebraic model: a polynomial function expressing its volume.
In this task, we translated the physical dimensions of a cylinder into an algebraic model: a polynomial function expressing its volume.
- This process included deriving equations for the radius and height and then substituting these back into the volume formula.
- After substituting, expanding polynomial expressions allowed us to refine the model into the final answer: \( V(x) = \pi (x^3 + 9x^2 + 24x + 20) \).
Other exercises in this chapter
Problem 78
For the following exercises, find the dimensions of the right circular cylinder described. The radius and height differ by one meter. The radius is larger and t
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For the following exercises, identify the removable discontinuity. $$ f(x)=\frac{2 x^{2}+5 x-3}{x+3} $$
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For the following exercises, find the dimensions of the right circular cylinder described. The radius and height differ by two meters. The height is greater and
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For the following exercises, identify the removable discontinuity. $$f(x)=\frac{x^{3}+x^{2}}{x+1}$$
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