# Finding equation for volume of rectangular Prism?

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.

• Sorry, I am new to this. I have updated the question to seem more appropriate. Thanks for your correction
– Ali
Apr 7, 2023 at 5:04
• 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.
– D S
Apr 7, 2023 at 5:18

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.