Question

Beans are sold in small or large cylindrical cans. The small can has a diameter of 4 inches and a height of 10 inches and costs \(3.50. The big can has a diameter of 8 inches and a height of 5 inches and costs \)6.50. Which is a better buy? (A) The small can (B) The large can (C) Both have the same value (D) Not enough information

Short answer

The large can is a better buy because it has a lower cost per unit volume (\(\frac{13}{160\pi}\)) compared to the small can (\(\frac{7}{80\pi}\)). Answer: (B) The large can

Step-by-step solution

Calculate the volume of the small can

To calculate the volume of the small can, we use the formula for the volume of a cylinder: \[V = \pi r^2 h\] where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height. The diameter of the small can is 4 inches, so the radius is 2 inches (\(\frac{4}{2}\)). The height is given as 10 inches. Plugging those values into the formula, we get: \[V_{small} = \pi (2)^2 (10) = 40\pi \text{ cubic inches}\]

Calculate the volume of the large can

Similarly, we'll use the same formula for the volume of the large can. The diameter of the large can is 8 inches, so the radius is 4 inches (\(\frac{8}{2}\)). The height is given as 5 inches. Plugging these values into the formula, we get: \[V_{large} = \pi (4)^2 (5) = 80\pi \text{ cubic inches}\]

Calculate the cost per unit volume for each can

Now that we have calculated the volumes, we will find the cost per unit volume for both cans by dividing their cost by their respective volume. For the small can: \[\text{Cost per unit volume}_{small} = \frac{\text{Cost}}{\text{Volume}} = \frac{3.50}{40\pi} = \frac{7}{80\pi} \text{ dollars per cubic inch}\] For the large can: \[\text{Cost per unit volume}_{large} = \frac{\text{Cost}}{\text{Volume}} = \frac{6.50}{80\pi} = \frac{13}{160\pi} \text{ dollars per cubic inch}\]

Compare the cost per unit volume

Now, we need to compare the cost per unit volume of each can to determine which is a better buy. \[\frac{7}{80\pi} \text{ (small can)} \quad \text{vs} \quad \frac{13}{160\pi} \text{ (large can)}\] \[160\pi\left(\frac{7}{80\pi}\right) = 160\pi\left(\frac{13}{160\pi}\right)\] \[7 \times 2 = 14 \neq 13\] Since \(14>13\), the cost per unit volume of the large can is less than that of the small can. Therefore, the large can is a better buy. Answer: (B) The large can

Key concepts

Cylinder Volume

Understanding how to calculate the volume of a cylinder is quite straightforward when you break it down into simple steps. A cylindrical shape, when viewed in math, is defined by three key measurements: the diameter, the radius, and the height. The radius, which is half the diameter, plays a crucial role in calculating the volume of a cylinder.

To find the volume, the formula used is \(V = \pi r^2 h\), where \(V\) is the volume, \(r\) is the radius, and \(h\) is the height. This formula essentially involves three steps:

• First, find the radius, which is half of the diameter.
• Second, square the radius \(r^2\) because the circular base of the cylinder needs to be multiplied by itself to get its area.
• Finally, multiply this area by the height \(h\) and \(\pi\).

So, if you have a cylinder with a diameter of 4 inches and a height of 10 inches, you start by calculating its radius as 2 inches (since \(\frac{4}{2} = 2\)).

Next, the area of the base is \((2^2)\) or 4 square inches, which you then multiply by the cylinder's height (10 inches) and \(\pi\), resulting in a volume of \(40\pi\) cubic inches. Understanding this calculation can help you in various problems involving volume, not just with cylinders, but with other shapes also.

Cost Per Unit Volume

Calculating the cost per unit volume is an important concept, especially when trying to determine the value of different package options. This concept involves two main steps: figuring out the total volume and then determining how much you're paying for each unit of that volume.

For each can, the cost per unit volume is derived by dividing the total cost of the product by its total volume. For example, if a small can of beans costs \(3.50\) with a total volume of \(40\pi\) cubic inches, the cost per unit volume is \(\frac{3.50}{40\pi}\), simplifying this gives \(\frac{7}{80\pi}\) dollars per cubic inch.

Similarly, for the large can costing \(6.50\) with a volume of \(80\pi\) cubic inches, the cost per unit volume comes out as \(\frac{6.50}{80\pi}\) or \(\frac{13}{160\pi}\) dollars per cubic inch.

Breaking costs down this way helps in understanding which option gives better value for the money spent. This metric is crucial for comparing products, such as food items packaged in different sizes, and selecting the one that maximizes your purchasing power.

Comparison of Values

When faced with two products that have varying sizes and costs, deciding which one offers better value requires comparing their cost per unit volume.

Once the cost per unit volume is calculated for each product, you will have two fractions that can be directly compared. For instance, comparing \(\frac{7}{80\pi}\) for the small can against \(\frac{13}{160\pi}\) for the large can requires simplifying the fractions or setting a common base to clearly see the difference.

Multiply each fraction by a common factor to remove \(\pi\) and make the denominators the same (if needed). Here, multiply the first by \(160\pi\) and the second by \(80\pi\), resulting in comparing \(7 \times 2\) against 13. Since \(14 > 13\), this calculation shows that the large can costs less per unit volume than the small can, making it the better value decision.

This skill is invaluable when making everyday purchasing decisions, as it ensures you choose the product that gives you more for your money.