Sure, let's solve the problem step by step, showing all the work.
### Part A:
Calvin builds a form for a cement step that is 36 inches long, 30 inches deep, and 6 inches high. We need to calculate the volume of the form to determine how much cement is needed to fill it.
The volume [tex]\( V \)[/tex] of a rectangular prism (which is the shape of this form) is calculated using the formula:
[tex]\[ V = \text{length} \times \text{depth} \times \text{height} \][/tex]
Given:
- Length = 36 inches
- Depth = 30 inches
- Height = 6 inches
Substituting these values into the formula:
[tex]\[ V = 36 \text{ inches} \times 30 \text{ inches} \times 6 \text{ inches} \][/tex]
Calculating the volume:
[tex]\[ V = 36 \times 30 \times 6 \][/tex]
[tex]\[ V = 1080 \][/tex]
[tex]\[ V = 6480 \text{ cubic inches} \][/tex]
So, the form requires 6480 cubic inches of cement to be completely filled.
### Part B:
Calvin has enough cement to fill 4500 cubic inches. He decides to change the length of the step so that he will use all the available cement. We need to find the new length.
We are given:
- Total cement available = 4500 cubic inches
- Depth = 30 inches
- Height = 6 inches
We will use the volume formula again but this time solve for the length [tex]\( L \)[/tex]:
[tex]\[ \text{Volume} = \text{length} \times \text{depth} \times \text{height} \][/tex]
[tex]\[ 4500 = L \times 30 \times 6 \][/tex]
Isolating [tex]\( L \)[/tex] (the new length):
[tex]\[ 4500 = L \times 180 \][/tex]
[tex]\[ L = \frac{4500}{180} \][/tex]
Calculating the new length:
[tex]\[ L = \frac{4500}{180} \][/tex]
[tex]\[ L = 25 \text{ inches} \][/tex]
Therefore, the new length of the step, so that Calvin will use all of the cement, is 25 inches.
