SOLUTION: The perimeter of the base of a crate is 4 meters. The height of the crate is 1 meter less than the width. a) write an expression (function) for the volume of the crate in terms

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Question 1054953: The perimeter of the base of a crate is 4 meters. The height of the crate is 1 meter less than the width.
a) write an expression (function) for the volume of the crate in terms of the width. Show the steps to how you arrived at this expression.
b) which values of the width give realistic volumes? What are the corresponding values of length and height for the values of the width that result in realistic volumes? Explain why all of the possible width values do not give realistic volumes.
c) find the dimension (width, length, height) that result in the maximum volume to the nearest tenth of a meter.
d) Finally, give the maximum volume to the nearest tenth of a cubic meter.
Found 2 solutions by josgarithmetic, MathTherapy:
Answer by josgarithmetic(39800) About Me  (Show Source):
You can put this solution on YOUR website!
Excuse me for removing this, since I made some as yet unfound mistake.

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The solution should have been kept instead of removing, after rechecking.
x for length of base
y for width of base
z for height
and given z=y-1 by description, perimeter giving x+y=2,

v volume becomes v=x%5E3-3x%5E2%2B2x=x%282-x%29%281-x%29
and important restrictions are system%28x%3E2%2Cy%3E1%2Cz%3E0%29.

dv%2Fdx=3x%5E2-6x%2B2=0 will give for length of base, x=1%2B-+sqrt%283%29%2F3, but this WILL NOT WORK. I also suggest looking at a graph of volume v against x. A zero at x=2 and then increase without bound to the right of x=2.

Answer by MathTherapy(10810) About Me  (Show Source):
You can put this solution on YOUR website!

The perimeter of the base of a crate is 4 meters. The height of the crate is 1 meter less than the width.
a) write an expression (function) for the volume of the crate in terms of the width. Show the steps to how you arrived at this expression.
b) which values of the width give realistic volumes? What are the corresponding values of length and height for the values of the width that result in realistic volumes? Explain why all of the possible width values do not give realistic volumes.
c) find the dimension (width, length, height) that result in the maximum volume to the nearest tenth of a meter.
d) Finally, give the maximum volume to the nearest tenth of a cubic meter.
I'm not going to answer all questions for you. You have to do something, don't you?

Let length be L, and width, W			

Then L + W = 2______L = 2 - W			

Height = W - 1			



Find the derivative and solve the resulting equation to get W, or width of 1.577350269, or 0.422649731. The LATTER value for width CANNOT WORK

since it'll result in a negative value for the height, and as you should know, this is a "no-no." Thereby, that value will not return a realistic

value for the height, hence, not a realistic value for the volume.

There, I have done a) and b) for you. You should be able to figure your way through the other 2.