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

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(39630) (Show Source): You can put this solution on YOUR website!
Excuse me for removing this, since I made some as yet unfound mistake.

--
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
and important restrictions are .

will give for length of base, , 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(10557) (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.
RELATED QUESTIONS
The surface area of a rectangular crate used to ship a sculpture is 672 square inches. If (answered by josgarithmetic)
A crate has a height of 6 feet. The sum of the areas of all 6 sides of the crate is 40... (answered by richwmiller)
Translate the following situation into an equation. Do not solve. �Suppose a crate of... (answered by ewatrrr)
An open-top, square bottom crate is made of two materials--the sides are made of a light... (answered by josgarithmetic)
Mr Tong has some guavas and papayas in two crates. In crate A, the number of guavas and... (answered by josmiceli)
A rectangular crate has the dimensions of 32.1 inches by 28.2 inches by 1 foot. What is... (answered by josh_jordan,MathTherapy)
Mr Tong has some guavas and papayas in two crates. In crate A, the number of guavas and... (answered by Theo)
Find the length of the diagonal of a side of a cubic packing crate whose volume is 2... (answered by Fombitz)
Find the length of a side of a cubic packing crate whose volume is 2 cubic... (answered by Fombitz)