Mrs. Duffy owns a restaurant supply store. She has 810 cubic inches of flour to store in her warehouse.

Question

Mrs. Duffy wants to store all of the flour in a container in the shape of a rectangular prism. The container should be the exact size to store all the flour without any extra room. What is one possible set of dimensions that she could use for the container? Give one combination of length, width, and height. Show or explain your work.

Mrs. Duffy completely fills two containers with flour. Each container is in the shape of a rectangular prism with a length of 5 inches, a width of 7 inches, and a height of 3 inches. She calculates the total volume of flour that she used as shown.

Mrs. Duffy checks her work and determines she made a mistake. Explain her error.

Mrs. Duffy will purchase one more container to hold another 810 cubic inches of flour. She is thinking of buying the container shown.

Is this container large enough to hold all 810 cubic inches of flour? Explain why or why not.

Answer 1

To help Mrs. Duffy with her container needs, let's go through each part of your question step by step.

Part 1: Possible Dimensions for the Container

Mrs. Duffy needs to store 810 cubic inches of flour in a container shaped like a rectangular prism, where the volume \(V\) is given by the formula:

\[ V = \text{length} \times \text{width} \times \text{height} \]

We need to find length, width, and height such that:

\[ \text{length} \times \text{width} \times \text{height} = 810 \]

One possible set of dimensions could be:

• Length = 10 inches
• Width = 9 inches
• Height = 9 inches

Calculating the volume:

\[ 10 \times 9 \times 9 = 810 \text{ cubic inches} \]

Thus, the dimensions of 10 inches (length), 9 inches (width), and 9 inches (height) will exactly hold 810 cubic inches of flour.

Part 2: Mrs. Duffy's Error in Volume Calculation

Mrs. Duffy filled two containers, each with dimensions length = 5 inches, width = 7 inches, height = 3 inches. The volume for one container can be calculated as follows:

\[ \text{Volume of one container} = 5 \times 7 \times 3 = 105 \text{ cubic inches} \]

She has two such containers, therefore:

\[ \text{Total volume} = 2 \times 105 = 210 \text{ cubic inches} \]

If Mrs. Duffy mistakenly calculated the total volume of flour used as something other than 210 cubic inches, that would be her error. The mistake might have been due to miscalculating the volume of each container or incorrectly adding up the volumes.

Part 3: Checking If the New Container is Large Enough

To determine if Mrs. Duffy's new container is large enough to hold all 810 cubic inches of flour, we'll need to calculate the volume of the container shown for her consideration.

Let’s say the potential dimensions of this new container are given as:

• Length = L (in inches)
• Width = W (in inches)
• Height = H (in inches)

The volume of the new container is calculated as:

\[ V = L \times W \times H \]

If the product of these dimensions equals or exceeds 810 cubic inches, then the container is large enough.

For example, if the new container is 12 inches long, 10 inches wide, and 7 inches high, the volume would be:

\[ 12 \times 10 \times 7 = 840 \text{ cubic inches} \]

Since 840 cubic inches is greater than 810 cubic inches, the container in this example would indeed be large enough to hold all her flour.

If you have specific dimensions for Mrs. Duffy's new container mentioned, please provide them to determine if it is sufficient or not.