Question

Soup is sold in both rectangular cardboard cartons and large cylindrical cans. The carton has dimensions of 3 inches by 4 inches by 7 inches and costs \(4.20. The can has a radius of 3 inches and a height of 4 inches and costs \)6.79. Which is a better buy? (A) The can is a better value. (B) The carton is a better value. (C) Both containers offer the same value. (D) There is not enough information to determine the answer.

Short answer

(B) The carton is a better value.

Step-by-step solution

Find the volume of the carton

To find the volume of the rectangular carton, use the formula for the volume of a rectangular prism: volume = length × width × height. The dimensions of the carton are 3 inches by 4 inches by 7 inches. Volume of carton = (3 inches) × (4 inches) × (7 inches) = 84 cubic inches

Find the volume of the can

To find the volume of the cylindrical can, use the formula for the volume of a cylinder: volume = π × (radius)^2 × height. The can has a radius of 3 inches and a height of 4 inches. Volume of can = π × (3 inches)^2 × (4 inches) = 36π cubic inches ≈ 113.1 cubic inches

Calculate the cost per unit volume for each container

Now that we have the volume for both the carton and the can, we can calculate the cost per unit volume for each by dividing the cost of each container by the respective volume. Cost per unit volume for the carton = (\(4.20) / (84) = 0.05 $/cubic inch\) Cost per unit volume for the can = (\(6.79) / (113.1) ≈ 0.06 $/cubic inch\)

Compare the cost per unit volume and determine the better value

The last step is to compare the cost per unit volume for the carton and the can. Carton: 0.05 $/cubic inch Can: 0.06 $/cubic inch Since the cost per unit volume of the carton is less than the cost per unit volume of the can, the carton is a better value. Therefore, the answer is (B) The carton is a better value.

Key concepts

Volume Calculation

Understanding how to calculate volume is essential in determining the capacity of different shapes and objects. Volume is the measure of how much space is occupied by a three-dimensional object, such as a carton or a can of soup. In our example, two different shapes were considered: a rectangular prism (the carton) and a cylinder (the can).

For a rectangular prism, the formula to find the volume is simply the product of its length, width, and height: \[ \text{Volume of rectangular prism} = \text{length} \times \text{width} \times \text{height} \].

For cylinders, the calculation is a bit different and involves \( \pi \) (Pi), a mathematical constant approximately equal to 3.14159. The volume of a cylinder is given by the formula: \[ \text{Volume of cylinder} = \pi \times \text{radius}^2 \times \text{height} \].

After calculating the volumes for both the carton and the can, it becomes clear how much space each container occupies and this is the first step in determining which package offers the better value.

Unit Price Comparison

Comparing unit prices is a crucial skill, especially when it comes to shopping and making decisions about value. The unit price tells us the cost for one unit of measure of a product, allowing for a fair comparison regardless of packaging or size differences.

To calculate the unit price, divide the total cost of the product by the total volume or quantity. In this case, we examined the cost per cubic inch of the soup containers to see which offered more soup for less money. The formulas would look like this: \[ \text{Unit Price} = \frac{\text{Total Cost}}{\text{Total Volume}} \].

Calculating Better Value

Once the unit prices for the carton and can are found, comparing the two gives a clear indication of which container gives more value for the money spent. If the unit price of one is lower than the other, that item is generally the better buy. Here, our carton, at \(0.05 per cubic inch, had a lower unit price than the can, which was \)0.06 per cubic inch, signaling that the carton is the more economical choice.

Geometry Math Problems

Geometry problems in math often involve the understanding and application of various shapes and their properties. These problems can range from simple calculations of area and volume to more complex scenarios that involve spatial reasoning and theorems.

In the case of our soup container example, we applied geometric concepts to solve a real-world problem. Working through geometry problems like these not only helps to solidify one's understanding of mathematical formulas but also improves problem-solving skills by requiring the application of these concepts in practical contexts.

Understanding the properties of different geometric shapes, like the rectangular prism and cylinder here, is fundamental. This knowledge is not just academic; it's practical and highly applicable in everyday life, from packing goods to purchasing products based on best value, just as we did in determining the better buy between the carton and the can of soup.