Dimensions that ive the most area

[Toxic Waste]

Hello everyone,

I almost done with all the homework from my calculus class, but I got stuck on one problem, only one problem, but very difficult to understand.
I would really appreciated if someone helps me with it. I don't even know how to start it : /
here it goes...

" Your neighbors are going to knock out the bottom of the entire length of the south wall of their house and turn it into a green house by replacing some bottom portion of the wall by a huge sloped piece of glass(which is expensive). They have already decided they are going to spend a certain fixed amout. The triangular ends of the greenhouse will be made of various materials they already have lying around.

The floor space in the greenhouse is only considered usable if they can both stand up in it, so part of it will be unusable, but they don't know how much. Of course this depends on how they configure the greenhouse. They want to choose the dimentions of the greenhouse to get the most usable floor space in it, but they are at a real loss to know wha the dimensions should be and how much usable space they will get. Fortunately, they know you are taking calculus. Amaze them!""

 

[jpr0]

Firstly I guess you can assume that the length of the piece of glass they choose, call it $l$, is determined by their budget. The width of the glass $w$ is fixed by the width of the wall. Now you have to place the glass against the wall so that it forms a right-angled triangle with the wall and the ground. You set up a cartesian coordinate system and let the wall lie on the line $x=0$, and the ground is then $y=0$.

The distance from the base of the wall to the part where the glass touches the wall, let's call it $y_0$, will be related to the distance from the base of the wall to the point on the ground where the glass touches the ground, which you can call $x_0$. This is a right angled triangle so you can relate $x_0$, $y_0$ and $l$. You can also derive and expression for the $y$ coordinate of any part of the glass as a function of the $x$ coordinate.

Let's call the height of the people $y_1$. Using the equation you just derived, you can work out the distance along $x$ in which the height of the glass it greater than their height, call it $x_1$. Once you have this, you can work out the total area in the greenhouse that is usable to them. Everything should be expressed in terms of $l$, $w$, $y_0$ and $y_1$. You then have to work out which choice of $y_0$ will maximize this area.

 

[Toxic Waste]

ok,
so far, I set up a cartsian coordinate system and derived the equation for the y coordinate( as explained in the second paragraph of jpr0 reply)
Yo= (L^2- Xo^2)^.5 that's the square root of L and Xo.

now, how can i work out the distance along in which the height of the glass it greater than their height?

 

[jpr0]

You have worked out the relationship between $y_0$, $l$ and $x_0$. Now you need an equation which tells you "for any distance (x) away from the wall, what is the height of the glass from the floor (y)". In other words you want an equation like:

$$y = -mx + c$$

so that for any distance away from the wall (x) you know at what height (y) the glass will be. In the equation above there are 2 undetermined constants, $m$ and $c$. You have to work these two constants out. Remember this is an equation which is describing the height of any part of the glass from the ground for any given distance from the wall. You already know 2 points on this line, right?

 

[Toxic Waste]

You have worked out the relationship between $y_0$, $l$ and $x_0$. Now you need an equation which tells you "for any distance (x) away from the wall, what is the height of the glass from the floor (y)". In other words you want an equation like:

$$y = -mx + c$$

so that for any distance away from the wall (x) you know at what height (y) the glass will be. In the equation above there are 2 undetermined constants, $m$ and $c$. You have to work these two constants out. Remember this is an equation which is describing the height of any part of the glass from the ground for any given distance from the wall. You already know 2 points on this line, right?

I don't get how i can figure out constants m and c in equation Y= mx+c if i don't have any numbers to work with.
Is m the slope of the glass on the wall?

 

[jpr0]

I don't get how i can figure out constants m and c in equation Y= mx+c if i don't have any numbers to work with.
Is m the slope of the glass on the wall?

y=mx+c describes the glass. The glass is touching the ground, and the wall. You know the coordinates at both of these points, because you have just worked them out. Each of these coordinates should satisfy the above equation, so choose m and c so that each point does satisfy the equation. And yes, m is the slope of the glass.