A piece of 8.5-by-11-inch cardboard has identical squares cut from its corners. It is then folded into a box with no lid. The volume of the box in cubic inches is 4c^3 - 39c^2 + 93.5c, where c is the side length of the missing squares in inches.
a. What is the volume of the box if c =1 in.?
A: 58.5?
b. What is the volume of the box if c = 4.25 in.?
A: 0.
c. Does your answer to part c make sense? Explain why or why not.
A: Yes, because it is solved by order of operations? (I am not entirely certain.)
a) correct
b) correct, but do you know why the volume is zero ?
c) this will also answer part b)
the given volume = c(8.5-2c)(11-2c) in expanded form
notice if c = 4.25, the size of the cut-out, you would be cutting away 4.25 on each side of the width
but the width is only 8.5.
So by cutting away 4.25 on each end of the 8.5, you would have no width left for the base of the box
the dimensions of the box would be
0 by (11 -2(4/25) by 4/25 which of course gives us a zero.
notice that in my factored form of the equation, there would be a restriction of
0 < c < 4.25
To find the volume of the box, we can start by determining the dimensions of the box after folding.
Step 1: Determine the length and width of the base of the box:
Given that the cardboard is 8.5 by 11 inches, we need to subtract 2c from both dimensions to account for the squares cut from the corners.
Length of the base = 11 - 2c inches
Width of the base = 8.5 - 2c inches
Step 2: Determine the height of the box:
The height of the box is equal to the side length of the missing squares, which is represented by c inches.
Step 3: Calculate the volume of the box:
The volume of a rectangular prism is calculated by multiplying the length, width, and height.
Volume = Length × Width × Height
Volume = (11 - 2c) × (8.5 - 2c) × c
Volume = 4c^3 - 39c^2 + 93.5c
a. To find the volume when c = 1 inch:
Substitute c = 1 into the equation:
Volume = 4(1)^3 - 39(1)^2 + 93.5(1)
Volume = 4 - 39 + 93.5
Volume = 58.5 cubic inches
b. To find the volume when c = 4.25 inches:
Substitute c = 4.25 into the equation:
Volume = 4(4.25)^3 - 39(4.25)^2 + 93.5(4.25)
Volume = 4(76.515625) - 39(18.0625) + 93.5(4.25)
Volume ≈ 306.0625 - 702.1875 + 396.875
Volume ≈ 0 cubic inches
c. Does your answer to part c make sense? Explain why or why not.
No, the answer does not make sense. The volume of the box cannot be zero because the dimensions of the box are positive values. It is possible that a mistake was made in the calculation or there may be an error in representing the equation.
To find the volume of the box in cubic inches, we need to substitute the given value of c into the volume equation.
a. If c = 1 inch, we can substitute this value into the volume equation:
Volume = 4c^3 - 39c^2 + 93.5c
= 4(1)^3 - 39(1)^2 + 93.5(1)
= 4 - 39 + 93.5
= 58.5
Therefore, the volume of the box when c = 1 inch is 58.5 cubic inches.
b. If c = 4.25 inches, we can substitute this value into the volume equation:
Volume = 4c^3 - 39c^2 + 93.5c
= 4(4.25)^3 - 39(4.25)^2 + 93.5(4.25)
= 4(76.265625) - 39(18.0625) + 93.5(4.25)
= 305.0625 - 703.125 + 397.375
= 0
Therefore, the volume of the box when c = 4.25 inches is 0 cubic inches.
c. When c = 4.25 inches, the volume of the box is 0 cubic inches. This means that the box cannot be formed because the length of the side of the missing squares is too large for the given dimensions of the cardboard.
To explain why the answer is 0, we can go back to the volume equation: 4c^3 - 39c^2 + 93.5c. We can see that when we substitute c = 4.25 inches, all three terms in the equation become 0. This happens because, with a side length of 4.25 inches for the missing squares, the cardboard cannot be folded into a box. Thus, the volume becomes 0, indicating that the box cannot exist.