On a rectangular piece of cardboard with perimeter 10 inches, three parallel and equally spaced creases are made (see Figure 1).
The cardboard is then folded along the creases to make a rectangular box with open ends (see Figure 2). Letting x represent the distance (in inches) between the creases, use the ALEKS graphing calculator to find the value of x that maximizes the volume enclosed by this box. Then give the maximum volume. Round your responses to two decimal places.
1 answer:
<h3>Answer:</h3>
- x = 5/6 in ≈ 0.83 in
- v = 125/108 in³ ≈ 1.16 in³
The width of the cardboard (in inches) will be 4x, and its length will be (5-4x) so the perimeter adds up to 10 in. Then the volume is the product of the area of the open ends (x²) and the length (5-4x).
The graphing calculator shows that volume is maximized when the value of x is 0.833, or 5/6 inch. The corresponding length is 5-4x = 5/3 inch, so the volume is ...
... v = (5/6)²×(5/3) = 125/108 . . . . in³
The corresponding decimal values of x and volume are ...
... x ≈ 0.83 in
... v ≈ 1.16 in³
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