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I have been stuck on this question for a while and would appreciate it if someone were to post a detailed step process of how they achieved an answer to part a

A closed rectangular box has length $x$ cm, width $y$ cm, and height $h$ cm. It is to be made from $300$ sq. cm of thin sheet metal, and the perimeter of the base is to be $40$ cm.

a) Show that the volume of the box is given by $V = 150h − 20h^2$.

b) Hence find the dimensions of the box that meets all the requirements and has the maximum possible volume.

I have come up with the following equations:

  1. $40 = 2x + 2y$ (perimeter of base)

  2. $300 = 2xh + 2xy + 2hy$ (equation for surface area)

I have attempted in making either $x$ or $y$ the subject in equation 1, and substituting it into equation 2, however always end up with $150 = -x^2 + 20x + 20h$ when trying to find the expression for volume. I'm unsure where I went wrong.

Thanks for your time

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  • $\begingroup$ Sorry, I am new to this. I have updated the question to seem more appropriate. Thanks for your correction $\endgroup$
    – Ali
    Apr 7, 2023 at 5:04
  • $\begingroup$ Hint: from the equation, you have marked as 2. , divide by 2 to get $150 = hx+hy+xy$. Then, $150 = h(x+y) + xy$. Which gives you $150 - 20h = xy$ using the equation marked as 1. Now, use the fact that $xyh = V$ to complete the proof. $\endgroup$
    – D S
    Apr 7, 2023 at 5:18

1 Answer 1

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Just to detail the hint in the comments:
You have already got $$2x+2y = 40$$ and $$2xh+ 2yh + 2xy = 300 $$ Divide both equations by $2$ to get: $$x+y = 20 \tag{1}\label{1}$$$$xh+yh+xy = 150\tag{2}\label{2}$$ Volume of a cuboid (or a rectangular prism) with dimensions $x,y,h$ is given by: $$V = xyh \iff xy = \frac Vh\tag{3}\label{3}$$ From $\eqref{2}$, $$150 = xh+yh+xy = h(x+y) + xy$$ Put the value of $x+y$ found in $\eqref{1}$ in this equation to get $$150 = 20h+xy$$ Put the value of $xy$ found in $\eqref{3}$ $$150 = 20h + \frac Vh$$ Multiplying by $h$ and rearranging the terms completes the proof.


We can also use the equation you arrived at: $$150 = 20h + 20x - x^2$$ $$150 = 20h + x(20 - x)$$ But, from $\eqref{1}$, $20 - x = y$ : $$150 = 20h+xy$$ Further steps are the same as in the above method.

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