Problem 88 - Algebra and Trigonometry

Question

Dimensions of a Can A cylindrical can has a volume of 40\(\pi \mathrm{cm}^{3}\) and is 10 \(\mathrm{cm}\) tall. What is its diameter? [Hint: Use the volume formula listed on the inside front cover of this book.]

Step-by-Step Solution

Answer

The diameter of the can is 4 cm.

Step 1: Understand the Volume Formula

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( V \) is the volume, \( r \) is the radius, and \( h \) is the height of the cylinder. In this problem, \( V = 40\pi \text{ cm}^3 \) and \( h = 10 \text{ cm} \).

Step 2: Set Up the Equation

Substitute the known values into the volume formula: \( 40\pi = \pi r^2 (10) \).

Step 3: Simplify the Equation

Divide both sides of the equation by \( \pi \) to get \( 40 = 10r^2 \).

Step 4: Solve for the Radius

Divide both sides of the equation by 10 to isolate \( r^2 \), giving \( r^2 = 4 \).

Step 5: Calculate the Radius

Take the square root of both sides to find \( r \), so \( r = 2 \text{ cm} \).

Step 6: Find the Diameter

The diameter of the cylinder is twice the radius, so the diameter is \( 2r = 4 \text{ cm} \).

Key Concepts

Cylinder Dimensions

Understanding the dimensions of a cylinder is crucial when dealing with problems related to its volume. A cylinder has two main components that define its shape:

• The height (h) - This is the straight-line distance between the two bases of the cylinder.
• The radius (r) - This is the distance from the center to the edge of the base, which is circular.

When given the volume of a cylinder, knowing these dimensions allows you to explore other related aspects such as the surface area or, in some cases, determine the unknown dimensions. In this exercise, we are particularly interested in finding the diameter, which is twice the radius. Thus, knowing either the radius or the height can set us on the path to solve for other dimensions effectively.

Mathematical Formulas

Mathematical formulas are the backbone of solving geometric problems, particularly when it involves calculating volumes or surface areas. For a cylinder, the volume formula is expressed as:\[ V = \pi r^2 h \] Here:

• \( V \) stands for volume.
• \( \pi \) is a constant approximately equal to 3.14159.
• \( r \) is the radius of the base.
• \( h \) is the height of the cylinder.

The formula encapsulates how the space within a vertical prism like a cylinder is quantified. Applying this formula to known values from the problem gives a clear pathway to find the unknown dimensions, such as the radius or diameter.

Problem Solving Steps

Problem solving in geometry involves a series of logical steps that guide you to the solution. Here's a detailed description using the cylinder exercise: First, understand the formula you're working with, which is crucial. Here, we use the formula for the volume of a cylinder. Our task further simplifies as we already know the height and volume.

Next, substitute the known values into the equation. By inputting our volume and height into the equation, we isolate the term we need to solve for - in this case, the radius.

The third step involves simplifying the equation to make calculations manageable. Observing common factors will aid in reducing the complexities, such as dividing both sides by constants to find essential values.

Once simplified, solve for the given variable. Here, deriving the radius involves simple arithmetic followed by square rooting the value obtained.

Finally, to find the diameter, a straightforward multiplication can be done once the radius is known, since the diameter is twice the radius. Breaking down the problem this way ensures clarity and accuracy at every successive step.