Problem 8 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. The volume \(V\) of a cylindrical can is modeled by the formula $$ V=\pi r^{2} h $$ where \(r\) is the radius and \(h\) is the height of the can. Find the volume of a can with radius 3 in. and height 5 in.

Step-by-Step Solution

Answer

The volume of the can is \(45\pi\) cubic inches.

Step 1: Identify Given Values

We have been provided with two essential dimensions of the cylindrical can: the radius \(r\) is 3 inches, and the height \(h\) is 5 inches.

Step 2: Understand the Formula

The volume \(V\) of the cylindrical can is given by the formula \(V=\pi r^2 h\). This formula requires the radius \(r\) and height \(h\) to calculate the volume.

Step 3: Substitute Values into the Formula

Insert the given values \(r = 3\) and \(h = 5\) into the formula: \(V = \pi (3)^2 (5)\).

Step 4: Calculate the Radius Squared

Compute \((3)^2\), which equals 9.

Step 5: Calculate the Volume

Multiply the squared radius (9) by the height (5), giving us \(45\). Then multiply by \(\pi\) to get the volume: \(V = 45\pi\).

Step 6: Express the Answer

The volume of the can is \(45\pi\) cubic inches.

Key Concepts

Cylindrical Can Calculation

Cylindrical cans are a common shape used for many practical items, like food cans or even some water bottles. Understanding how to calculate the volume of a cylindrical can is useful for various everyday applications.
When we talk about calculating the volume of a cylinder, we mean finding out the amount of space inside the cylinder. Volume is measured in cubic units, such as cubic inches, cubic centimeters, or cubic meters. This helps us understand how much material or liquid a can can hold.
Because of its symmetrical shape, calculating the volume of a cylindrical can only requires knowing two measurements: the radius and the height. Once we have these, we can use simple mathematics to find out the volume.

Radius and Height

When dealing with a cylindrical can, two key measurements are necessary to calculate its volume: the radius and the height.

• Radius ( ext{ }) : This is the distance from the center of the circular base to the edge (or rim). For example, if a cylinder's base looks like a large circle, the radius is half the diameter of this circle.
• Height ( h ext{ }) : This is the distance from the bottom of the cylinder to the top. Essentially, it measures how tall the cylinder is.

These two measurements are crucial because they are inputs in the mathematical formula that calculates the volume. Incorrectly measuring or understanding these can result in miscalculations of the cylinder's capacity.

Mathematical Formula for Volume

The mathematical formula to find the volume of a cylinder combines both the radius and the height.This formula is: \[ V = \pi r^2 h \]Let's break it down:

• \( \pi \): Pi is a constant in mathematics, approximately equal to 3.14159, representing the ratio of the circumference of a circle to its diameter.
• \( r^2 \): This part of the formula means the radius squared, or the radius multiplied by itself.
• \( h \): Multiplying by the height stacks up the circular bases through the whole height of the can, filling up its entire volume.

This formula combines the area of the base (\( \pi r^2 \)) with the height, effectively filling up the cylinder's "space" with its "base area" for its complete height.

Step-by-Step Solution

Solving the problem of finding the volume starts with identifying what you know and using the formula correctly. Here is a simple step-by-step approach:Step 1: Identify Given Values
First, confirm the known values of the can's size:

• Radius ( = 3 ext{ in.})
• Height ( h = 5 ext{ in.})

Step 2: Understand the Formula
Recall the formula: \( V = \pi r^2 h \). Both radius and height are needed.Step 3: Substitute Values
Plug in the known values: \( V = \pi (3)^2 (5) \).Step 4: Calculate the Radius Squared
Here, calculate \((3)^2 = 9\).Step 5: Calculate the Volume
Multiply your result: \( 9 \times 5 = 45 \), then \( 45 \times \pi = 45\pi \).Step 6: Express the Answer
The volume of the cylindrical can is \( 45\pi \) cubic inches. This means the can can hold this amount of space, illustrating how the circle's area repeated over the cylinder's height equals its volume.