Problem 63 - College Algebra with Modeling and Visualization

Question

The volume of a cylinder is given by \(V=\pi r^{2} h,\) where \(r\) is the radius and \(h\) is the height. If the height of a cylindrical can is 6 inches and the volume must be between \(24 \pi\) and \(54 \pi\) cubic inches, inclusive, find the possible values for the radius of the can

Step-by-Step Solution

Answer

The radius \( r \) must be between 2 and 3 inches, inclusive.

Step 1: Understand the problem statement

The volume of a cylinder is given by the formula \( V = \pi r^2 h \). We need to find the values of the radius \( r \) such that the volume \( V \) is between \( 24\pi \) and \( 54\pi \), inclusive, for a fixed height \( h = 6 \) inches.

Step 2: Set up the inequality for volume

Using the formula \( V = \pi r^2 h \) and substituting \( h = 6 \), we get \( V = \pi r^2 \times 6 = 6\pi r^2 \). The inequality for volume becomes \( 24\pi \leq 6\pi r^2 \leq 54\pi \).

Step 3: Simplify the inequality

Divide the entire inequality \( 24\pi \leq 6\pi r^2 \leq 54\pi \) by \( 6\pi \) to simplify it. This yields \( 4 \leq r^2 \leq 9 \).

Step 4: Solve for the radius

To find the possible values of \( r \), solve each part of the inequality \( 4 \leq r^2 \leq 9 \). The solutions for \( r^2 \) give \( r \leq \sqrt{9} = 3 \) and \( r \geq \sqrt{4} = 2 \). Thus, the radius \( r \) must satisfy \( 2 \leq r \leq 3 \).

Key Concepts

Volume Formula

One of the primary formulas every math student should know is the volume formula for a cylinder. This formula helps us find how much space a cylinder occupies. It's mathematically represented as \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder. This equation is vital in multiple practical applications, such as determining how much soup a can holds or the capacity of a tank. To understand it simply:

• \( \pi \) approximately equals 3.14159 and is a constant.
• \( r^2 \) roughly is the area of the circular base since it's \( \pi r^2 \).
• Multiplying the base area by the height \( h \) gives the volume.

If our cylinder's height is constant, then the volume varies with the square of the radius. It's a direct application of geometry used in real-world scenarios like engineering and construction.

Inequality Solving

Inequality solving is an essential mathematical skill that helps determine a range for an unknown variable. In this exercise, our goal was to find values of the radius that make the cylinder's volume fall within a predefined range. Given the inequality \(24\pi \leq 6\pi r^2 \leq 54\pi\), the steps are straightforward:

• Eliminate \(\pi\) by dividing all parts of the inequality by \(\pi\), simplifying calculations.
• Then simplify it to \(4 \leq r^2 \leq 9\).

Solving this inequality involves finding the value of \(r^2\) and then applying square roots on both ends of the inequality. This process tells us that \(r\) must be between 2 and 3, inclusive. Understanding inequality solving not only aids in mathematical problems but also in making decisions where conditions need to be met.

Radius Calculation

Calculating the radius of a cylinder often involves using algebra to manipulate a formula or inequality involving the radius itself. For this exercise, we aimed to find acceptable values for the radius given the cylinder's volume constraints.To solve the inequality \(4 \leq r^2 \leq 9\):

• Take the square root of all parts to resolve \(r^2\) into \(r\), giving \( r \geq \sqrt{4} \) and \( r \leq \sqrt{9} \).
• This results in \( r \geq 2 \) and \( r \leq 3 \).

This simplifies to \( 2 \leq r \leq 3 \), meaning our radius must be greater than or equal to 2 inches and less than or equal to 3 inches. Calculating the radius in this manner is a common task in problems where dimensions need to adhere to certain conditions, common in fields like manufacturing and product design.

Mathematical Modeling

Mathematical modeling involves using mathematics to represent, analyze, predict, or otherwise provide insight into real-world phenomena. In this exercise, mathematical modeling lets us understand the relationship between the cylinder’s volume and its radius. By framing this word problem with math, we:

• Translate physical conditions (such as volume constraints) into mathematical terms (inequalities).
• Apply formulas to determine feasible dimensions for real objects.

The initial inequality \( V = \pi r^2 h \) being set in the range \( 24\pi \leq 6\pi r^2 \leq 54\pi \), reflects an upper and lower bound on what our volume should be. Mathematical modeling is a critical skill, allowing students and professionals to tackle complex problems by breaking them down into manageable steps, using equations to gain insights.