Problem 57 - Algebra and Trigonometry with Analytic Geometry

Question

A manufacturer of tin cans wishes to construct a right circular cylindrical can of height 20 centimeters and capacity \(3000 \mathrm{~cm}^{3}\) (see the figure). Find the inner radius \(r\) of the can.

Step-by-Step Solution

Answer

The radius of the can is approximately 6.91 cm.

Step 1: Understand the Problem

We are given a cylindrical can with a height of 20 centimeters and a volume (capacity) of 3000 cm³. We need to find the radius of the base of the cylinder.

Step 2: Formula for Volume of a Cylinder

The formula for the volume of a cylinder is given by \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height. In this problem, \( V = 3000 \mathrm{~cm}^{3} \) and \( h = 20 \mathrm{~cm} \).

Step 3: Plug in the Known Values

Substitute the given values into the volume formula: \( 3000 = \pi r^2 (20) \).

Step 4: Solve for \( r^2 \)

First, divide both sides of the equation by 20 to isolate \( r^2 \) and resolve the equation: \( r^2 = \frac{3000}{20\pi} \). This simplifies to \( r^2 = \frac{150}{\pi} \).

Step 5: Find \( r \)

Take the square root of both sides: \( r = \sqrt{\frac{150}{\pi}} \). Using a calculator, we find \( r \approx 6.91 \mathrm{~cm} \).

Key Concepts

Cylinder Geometry

Cylinders are three-dimensional geometric shapes with two parallel circles as their bases. They have a straight side connecting these circular bases, forming the height of the cylinder. Understanding cylinder geometry is key to solving many mathematical problems that involve calculating volume or surface area. In the context of a tin can, we deal with a right circular cylinder, meaning the axis (height) is perpendicular to the bases.
Key features of a cylinder include:

• Circular bases: These define the width (or radius) of the cylinder.
• Height: This is the perpendicular distance between the bases.

The height and the radius are crucial when calculating other properties of the cylinder, such as volume or surface area.

Volume Formula

The volume of a cylinder tells us how much space is enclosed within. It is calculated using the formula:\[ V = \pi r^2 h \]where \( V \) represents the volume, \( r \) is the radius of the circular base, and \( h \) is the height.
In our exercise, the volume is given as 3000 cm³, and the height is 20 cm. By plugging these values into the formula, we gain an essential setup for finding unknown variables, like the radius in this case.
Learning this formula is important in geometry as it applies to many real-world problems involving cylindrical shapes, such as containers, pipes, and other objects.

Radius Calculation

Calculating the radius of a cylinder when volume and height are known involves working with the volume formula. We set the known values into the equation and solve for the unknown variable:1. Start with the formula: \( V = \pi r^2 h \)2. Insert the known values: \( 3000 = \pi r^2 \times 20 \)3. Simplify to isolate \( r^2 \): \[ r^2 = \frac{3000}{20\pi} \]4. Simplify further: \[ r^2 = \frac{150}{\pi} \]5. Finally, calculate \( r \) by taking the square root: \[ r = \sqrt{\frac{150}{\pi}} \approx 6.91 \text{ cm} \]These steps are systematic and allow you to solve for the radius methodically, ensuring consistent results every time.

Mathematics Problem Solving

Solving problems in mathematics often involves breaking down a problem into smaller and more manageable steps. This is especially the case with geometry problems.
Let's follow the steps used in the tin can problem:

• Understand the problem: Identify what is given and what is required.
• Formulate a plan: Use known formulas and relationships. For cylinders, this often means applying the volume formula \( V = \pi r^2 h \).
• Carry out the plan: Substitute known values and solve for unknowns step by step.
• Review and interpret: Once you have a solution, review your calculations and ensure they make sense in the context of the problem.

Applying this structured approach can simplify even complex problems and enhance understanding, building valuable skills for tackling all mathematics challenges.