Problem 46
Question
Two cylindrical cans of soup sell for the same price. One can has a diameter of 6 inches and a height of 5 inches. The other has a diameter of 5 inches and a height of 6 inches. Which can contains more soup and, therefore, is the better buy?
Step-by-Step Solution
Verified Answer
Can 1, which has a diameter of 6 inches and a height of 5 inches, contains more soup and is therefore the better buy.
1Step 1: Calculate the Radius
Divide the diameter by 2 to find the radius for each can as the radius is half of the diameter. For can 1, the radius is 3 inches and for can 2, the radius is 2.5 inches.
2Step 2: Calculate the Volume of Each Can
Substitute the radius and the height of each can into the volume formula \( \pi r^2h \). For Can 1, the volume is \( \pi * 3^2 * 5 = 45\pi \) cubic inches. For Can 2, the volume is \( \pi * 2.5^2 * 6 = 37.5\pi \) cubic inches.
3Step 3: Compare the Volumes
Since \( 45\pi > 37.5\pi \), Can 1 contains more soup.
Key Concepts
Understanding Cylindrical VolumeApplying Geometry ConceptsComparison of Volumes
Understanding Cylindrical Volume
Calculating the volume of a cylinder is an essential skill in geometry. Cylinders are three-dimensional shapes with a circular base and straight, parallel sides. To find the volume, you need two measurements: the radius of the circular base and the height of the cylinder.
The formula for the volume of a cylinder is given by \( V = \pi r^2 h \). Here, \( V \) stands for volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
Understanding this formula is crucial for solving real-world problems like determining which can of soup holds more. In our exercise, by applying \( \pi r^2 h \), we calculated the volumes of the two soup cans. Each can used the same operation, demonstrating how small changes in dimensions impact the overall volume.
The formula for the volume of a cylinder is given by \( V = \pi r^2 h \). Here, \( V \) stands for volume, \( r \) is the radius of the base, and \( h \) is the height of the cylinder.
Understanding this formula is crucial for solving real-world problems like determining which can of soup holds more. In our exercise, by applying \( \pi r^2 h \), we calculated the volumes of the two soup cans. Each can used the same operation, demonstrating how small changes in dimensions impact the overall volume.
Applying Geometry Concepts
Geometry is all about understanding shapes, sizes, and the relative positions of figures. When handling cylinders or any geometric shape, it's important to recognize relationships. These relationships help us understand how changes in dimensions affect overall properties like volume.
In the case of our soup cans, understanding that the diameter directly affects the radius is key. This is because the radius is half the diameter, and small changes in the radius can lead to significant differences in volume. Geometry encourages us to carefully consider the role each dimension plays in these calculations.
In the case of our soup cans, understanding that the diameter directly affects the radius is key. This is because the radius is half the diameter, and small changes in the radius can lead to significant differences in volume. Geometry encourages us to carefully consider the role each dimension plays in these calculations.
Comparison of Volumes
Comparing volumes is about assessing the amount of space different shapes occupy. It's incredibly useful in deciding which option is better for scenarios like purchasing or manufacturing.
To compare the volumes of the soup cans, we look at the derived volumes: Can 1 with a volume of \( 45\pi \) cubic inches and Can 2 with \( 37.5\pi \) cubic inches. Clearly, since \( 45\pi \) is greater than \( 37.5\pi \), Can 1 holds more soup.
To compare the volumes of the soup cans, we look at the derived volumes: Can 1 with a volume of \( 45\pi \) cubic inches and Can 2 with \( 37.5\pi \) cubic inches. Clearly, since \( 45\pi \) is greater than \( 37.5\pi \), Can 1 holds more soup.
- This comparison is straightforward because all other factors like price are constant.
- The larger volume means Can 1 offers more soup for the same price, making it the better buy.
Other exercises in this chapter
Problem 46
This activity is suggested for two or three people. Research some of the practical applications of topology. Present the results of your research in a seminar t
View solution Problem 46
A police helicopter is flying at 800 feet. A stolen car is sighted at an angle of depression of \(72^{\circ}\). Find the distance of the stolen car, to the near
View solution Problem 46
Use the Pythagorean Theorem to solve. Use your calculator to find square roots, rounding, if necessary, to the nearest tenth. This problem appeared on a high sc
View solution Problem 47
Research non-Euclidean geometry and plan a seminar based on your group's research. Each group member should research one of the following five areas: a. Present
View solution