To find one possible set of dimensions for Mrs. Duffy's container that can hold 810 cubic inches of flour in the shape of a rectangular prism, we can start by understanding that the volume \( V \) of a rectangular prism is given by the formula:
\[ V = \text{length} \times \text{width} \times \text{height} \]
We need to find integers \( l \), \( w \), and \( h \) such that:
\[ l \times w \times h = 810 \]
One set of dimensions that works is:
- Length (l) = 10 inches
- Width (w) = 9 inches
- Height (h) = 9 inches
Calculating the volume:
\[ 10 \times 9 \times 9 = 810 \text{ cubic inches} \]
So, one possible set of dimensions is 10 inches (length), 9 inches (width), and 9 inches (height).
Next, let's check Mrs. Duffy's calculation for the total volume of flour when she fills two containers, each with dimensions of length 5 inches, width 7 inches, and height 3 inches.
First, we calculate the volume of one container:
\[ \text{Volume of one container} = \text{length} \times \text{width} \times \text{height} = 5 \times 7 \times 3 \]
Calculating:
\[ 5 \times 7 = 35 \] \[ 35 \times 3 = 105 \text{ cubic inches} \]
Since she has two containers, the total volume of flour in both containers is:
\[ \text{Total volume} = 2 \times 105 = 210 \text{ cubic inches} \]
Mrs. Duffy's mistake comes from misunderstanding the total amount of flour. She originally thought she had stored 810 cubic inches of flour, but in fact, the total volume of flour in the two containers combined is only 210 cubic inches, which is significantly less than 810 cubic inches. Therefore, she miscalculated the flour volume she actually had when she claimed to fill the two containers.
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