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.

Answer 1

To find one possible set of dimensions for a rectangular prism with a volume of 810 cubic inches, we need to find three positive integers \( l \), \( w \), and \( h \) such that:

\[ l \times w \times h = 810 \]

We'll start by finding the prime factorization of 810:

\[ 810 = 2 \times 3^4 \times 5 \]

Next, we can choose factors of 810 to find suitable dimensions.

To simplify our choices, let's begin by considering one factor for length \( l \), another for width \( w \), and the remaining factor will be the height \( h \).

Let’s try the following factors:

• Let \( l = 10 \)
• Then we calculate \( \frac{810}{10} = 81 \).

Now, let’s find two dimensions \( w \) and \( h \) such that:

\[ w \times h = 81 \]

One straightforward choice is:

• Let \( w = 9 \) and \( h = 9 \) (since \( 9 \times 9 = 81 \)).

Therefore, one set of dimensions is:

• Length \( l = 10 \)
• Width \( w = 9 \)
• Height \( h = 9 \)

Now, let's verify:

\[ l \times w \times h = 10 \times 9 \times 9 = 10 \times 81 = 810 \]

This matches the required volume.

Thus, one possible set of dimensions for Mrs. Duffy’s container is:

Length: 10 inches, Width: 9 inches, Height: 9 inches.