Problem 301 - Precalculus

Question

For the following exercises, write the polynomial function that models the given situation. A right circular cone has a radius of \(3 x+6\) and a height 3 units less. Express the volume of the cone as a polynomial function. The volume of a cone is \(V=\frac{1}{3} \pi r^{2} h\) for radius \(r\) and height \(h\)

Step-by-Step Solution

Answer

The volume as a polynomial function is \( V(x) = \pi (9x^3 + 45x^2 + 72x + 36) \).

Step 1: Identify Radius and Height

The radius of the cone is given as \( r = 3x + 6 \). The height is 3 units less than the radius, so the height is \( h = (3x + 6) - 3 = 3x + 3 \).

Step 2: Substitute Values into Volume Formula

The formula for the volume of a cone is \( V = \frac{1}{3} \pi r^2 h \). Substitute \( r = 3x + 6 \) and \( h = 3x + 3 \) into the formula: \[ V = \frac{1}{3} \pi (3x + 6)^2 (3x + 3) \].

Step 3: Expand \( r^2 \)

Expand \((3x + 6)^2\) using the binomial square formula: \((a + b)^2 = a^2 + 2ab + b^2\). So, \[ (3x + 6)^2 = 9x^2 + 36x + 36 \].

Step 4: Multiply by Height

Multiply the expanded radius squared by the height: \[ (9x^2 + 36x + 36)(3x + 3) \]. Distribute each term of \( 3x + 3 \): \[ 9x^2(3x + 3) + 36x(3x + 3) + 36(3x + 3) \].

Step 5: Simplify Each Term

Calculate each component: \(9x^2(3x + 3) = 27x^3 + 27x^2\), \(36x(3x + 3) = 108x^2 + 108x\), and \(36(3x + 3) = 108x + 108\).

Step 6: Combine Like Terms

Combine the terms: \[ 27x^3 + (27x^2 + 108x^2) + (108x + 108x) + 108 = 27x^3 + 135x^2 + 216x + 108 \].

Step 7: Multiply by \(\frac{1}{3} \pi\)

Finally, incorporate the \(\frac{1}{3} \pi\) factor: \[ V = \frac{1}{3} \pi (27x^3 + 135x^2 + 216x + 108) \].

Step 8: Simplify the Expression

Divide each term by 3: \[ V = \pi (9x^3 + 45x^2 + 72x + 36) \]. The polynomial function for the volume is \( V(x) = \pi (9x^3 + 45x^2 + 72x + 36) \).

Key Concepts

Volume of a Cone

The volume of a cone is an essential concept in geometry that describes how much space is inside the cone. To find the volume of a cone, we use the formula: \[ V = \frac{1}{3} \pi r^2 h \]where:

• \( V \) is the volume
• \( r \) is the radius of the circular base
• \( h \) is the height of the cone

This formula implies that the volume of the cone is proportional to the product of the square of its radius and its height, all scaled by a factor of \( \frac{1}{3} \pi \). The factor of one-third accounts for the difference between a cone's volume and the volume of a cylinder with the same base and height. Each step in finding the volume involves determining these dimensions accurately and substituting them into the formula.

Radius and Height

Understanding the dimensions of the cone is crucial to applying the volume formula correctly. In the context of this problem, the radius is given as an algebraic expression, \( r = 3x + 6 \). The height is also given in terms of the radius. It is described as 3 units less than the radius, so mathematically we express it as \( h = (3x + 6) - 3 = 3x + 3 \).
This way of relating dimensions helps us model real-world problems where changes in one dimension affect another. Knowing how to manipulate these algebraic expressions to find the required measurements simplifies the process of solving geometric problems involving cones.

Binomial Expansion

When given an expression like \((3x + 6)^2\), we use binomial expansion to simplify it. This technique is derived from the formula: \[ (a + b)^2 = a^2 + 2ab + b^2 \]In our case, \( a = 3x \) and \( b = 6 \). So, applying the formula, we find:

• \( (3x + 6)^2 = 9x^2 + 36x + 36 \)

This step transforms the expression for the radius squared, making it easier to multiply later. Binomial expansion is a vital skill in algebra, allowing us to expand and simplify expressions for effective problem solving in various scenarios.

Multiplication and Simplification

Once we expand the binomial expression, the next steps involve multiplication and simplification. We multiply the expanded form of \( r^2 \) by the expression for the height: \[ (9x^2 + 36x + 36)(3x + 3) \]This requires distributing each term:

• \( 9x^2(3x + 3) = 27x^3 + 27x^2 \)
• \( 36x(3x + 3) = 108x^2 + 108x \)
• \( 36(3x + 3) = 108x + 108 \)

Finally, we combine like terms to create a simpler polynomial expression: \[ 27x^3 + 135x^2 + 216x + 108 \]After that, we incorporate the \( \pi \) factor and divide each term by 3 to finalize the volume function: \[ V(x) = \pi (9x^3 + 45x^2 + 72x + 36) \]Understanding this process is crucial, as it seamlessly combines expansion, multiplication, and simplification to arrive at a meaningful and usable polynomial function.