Find the height of the prism if the length is 9 inches, the width is 5 inches, and the volume of the prism is 315 cubic inches.
1 answer:
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Answer:
Height: 3/2 inches
Length: 12 inches
Width: 4 inches
Step-by-step explanation:
Let x is the side length of the square
The height of the box by cutting squares off :x
- The new length of the cardboard = 15 -2x (because we cut from 4 corners)
- The new width of the cardboard = 7 -2x (because we cut from 4 corners)
The new volume of it is:
V = (15 -2x) (7 -2x) x
<=> V =
To maximum volume, we use the first derivative of the volume
<=>
<=>
<=> 2x -3 = 0 or 6x -35 = 0
<=> x = 3/2 or x = 35/6
To determine which value of x gives a maximum, we evaluate
= 24x -88
- If x = 3/2, we have:
= 24(3/2) -88 = -52
- If x = 35/6, we have:
= 24(35/6) -88 = 52
We choose x = 3/2 to have the maximum volume because the value of x that gives a negative value is maximum.
So the dimensions (in inches) of the box is:
Height: 3/2 inches
Length: 15-2(3/2) = 12 inches
Width: 7 - 2(3/2) = 4 inches