Dear Al,

I received your letter and I have been working on your greenhouse problem. You asked me to help you figure out how long of a concrete block (extended from the side of your house) you would need to serve as the base of your greenhouse. The length of this concrete block will also be the length of floor space you will have for your greenhouse. In your letter, you told me that your brother had given you a piece of glass that measures 15 feet by 12 feet wide, and you also stated your concern that, since are 6 feet tall, you tend to hit your head when you don't have enough head room. So what you asked was for me to figure out where we can place the pane of glass (measured in feet away from your house) so you will have enough headroom and also the maximum amount of room to walk around in your greenhouse. And from this value I will be able to figure out how much concrete you will need to pour.

I just want to let you know that before I solved this problem, I made some assumptions: I assumed that the glass would not bend and will always stay straight. I also assumed that since you are 6 feet tall, you would need only that much room in the height of the greenhouse so you wont hit your head; I did NOT assume that you would need a couple of extra inches so you would not hit your head.

Now back to your problem, I found that you would need to extend the concrete block 10 feet from the base of your house, so when the people come to pour the concrete today, they should measure 10 feet from the base of your house (where you want the greenhouse) and to pour it there, until they reach 10 feet from the base of your house. And the width of the concrete along the side of your house will be 12 feet wide, because that is the width of the glass. So you will have a floor space of 10 feet by 12 feet wide inside your greenhouse. Here is a picture of what your concrete block will look like in comparison to your house:

Now I am going to draw you a diagram, like the one you drew me in your letter, to show you with pictures how I started to solve your greenhouse problem:

Now I will explain all the variables that I used in this picture. The variable B (at the bottom of the picture) represents the length in feet of the concrete block, extended from the side of your house. This is what you asked me to find out so you would know how much concrete you would need to pour today. The variable H represents the height in feet of where the pane of glass is placed on your house. So pane of glass will rest on your house "H" feet high from the base of your house. The value "15 feet" represents the length of the glass that you have for your greenhouse. The value "6 feet" represents your height. This was used in the problem so you would have enough headroom to walk around without hitting your head, and so it will be figured into the final answer. The variable X represents how much room you will have to walk around without hitting your head. Just a little note, when I solved this problem I wanted to give you the most area possible to walk around in your greenhouse, without hitting your head, given the piece of glass that you have. So I made "X" the maximum it can be in this case. The last variable is "H-6" and this is the area above your head. I got this value from taking your height and subtracting it from the height of where the glass will sit on your house. So if you look at the picture, the value of "H-6" is the height of space above your head. This should explain all the variables and values that will be used to solve your greenhouse problem.

Now I will explain where I got my equations from the picture to solve your problem. First, you told me that the pane of glass measures 15 by 12 feet wide. We are going to have the side of the glass that is 12 feet wide sitting on the ground (and I will explain why we did NOT put the side that measures 15 feet on the ground). From the picture, you can see that your greenhouse is one big triangle:

So, if you remember some high school math, the equation for the Pythagorean theorem will create one equation for our greenhouse problem. The Pythagorean theorem says, (when we have right triangles, which we do, because the house and the ground make a 90-degree angle), that: a + b = c . To fit this equation to our problem, the equation changes to:

Or, to put this equation in terms of B, because this is the variable we want to solve for eventually because it represents the length in feet of the concrete that needs to be laid and also how far out in feet to lean the glass to the house, the equation (which is still the Pythagorean theorem) changes to this:

Since there are two unknown values (for B and for H) we need more information to solve this problem. If we look at the picture again, we can see that there are actually two triangles, and not just the one we used above. I will draw you another picture to show you where the second triangle is, it will be shaded so it can be quickly identified:

These two triangles, if you remember some high school geometry, are similar, they both have a 90-degree angle. But, since they are similar, we can get another equation. This equation (solved for X, because that is the variable that we want to maximize the value of) looks like this:

If you look back to the first equation we found, it is solved for the variable B, so we can substitute the other equation into the new one we just found anywhere we see the variable B. So the new equation we found will now look like this:

Before using this equation (and some calculus) to solve your problem, I will show you graphically the answer to your problem. Now, like you said in your letter, if we were to place the glass on your house at the value H = 6 feet (remember H represents the variable of the height in feet of where we place your glass, and 6 feet represents your height in feet), you would have no room to walk, because the glass would slope down, but you would have a lot of room to put plants. Unfortunately you wouldn't be able to tend to them. Also, like you said, if we put the glass at H = 15 (which is the height of the pane of glass you have), you would have a lot of head room, but no place to put your plants, and even for you to fit in there, because the glass would be vertical. So we want to find the point where H = some value where you will have the maximum space to walk around (without hitting your head), which is represented by the variable X as seen in the picture, and also the most floor space (or longest base (B) of concrete) to correspond to that value. After doing the calculation (which I will explain a little further down, now I will show it graphically) I learned that the glass should be placed on the house at H = 11 feet, because this is the value which makes our variable X (room for you to walk in the greenhouse without hitting your head) a maximum. Here is a graph that compares the value we want for H (which is 11 feet) to the other options (where H = 6 feet or where H = 15 feet):

Now I will show you how I found this value (H = 11 feet) with calculus. The way in calculus to maximize a variable, like we want to do to X (so you will have a lot of room to walk around without hitting your head), is to take the derivative of the equation, then set that new equation equal to zero, and finally solve this equation. In calculus a derivative can also be called a slope, if that explanation helps you to understand the term. If you remember some calculus, the derivative of our final equation (X' represents the derivative equation) we found (see above) looks like this:

When this equation is set equal to zero and solved, we will get values for our variable H that will correspond to either a maximum or minimum value for our variable X. So after I set this equation equal to zero and solved, I found that our X variable is at a maximum when H = 11 feet.

Now that we know the value for one of our variables, we can plug this value into the equation that was set equal to X (before we took the derivative of this equation!!). So the equation with the value 11 (feet) plugged in for the variable H, now looks like this:

When we solve for X with this equation, we learn that X = 4.5 feet. So this means you will have 4.5 feet to walk, without hitting your head, if you are walking away from your house and towards the pane of glass.

I am going to take a minute, as I promised much earlier in the letter, to explain why is it better to place the glass with the 12 feet wide side on the ground. So just to show you that the way I have been solving the problem is right, we are going to switch the side of the glass so now the side which measures 15 feet will be the side that is placed on the ground. Just a warning, this is not the best way to solve the problem, I am just putting it in to show you that it is better to place the glass the way we originally did, with the side of glass that measures 12 feet on the ground. The value of 15 feet, for the side of the glass that we used earlier in the equations, is only seen in simple form in the derivative equation (the equation we used to solve for the H value, where X would be a max). So we just need to plug in the value for 12 , which is 144, where we see 15², or 225. The derivative equation, if we were going to place the glass so the side that measured 15 feet was on the ground, now would look like this:

When we use the same technique as above to find the value for H which makes the value for X a max, we learn that H = 9.5 feet. Also, like I did before I am going to plug this value into the equation, before the derivative was taken. In this case, we get a value for X = 2 feet. So in this case you would only have two feet to walk, without hitting your head, from your house to the edge of the glass. Now that you know I was solving the your problem properly, I will return to solving it.

To refresh your memory with the values I found before, the value for the variable H = 11 feet, so you will place the piece of glass at the point that is 11 feet high on your house. And we also found the value for the variable X; X = 4.5 feet. So you will have 4.5 feet to walk, without hitting your head, from your house to the end of the greenhouse. Now that we know these to variables, we can solve for the variable B, which is the measure in feet from you house to where you should place the glass, and also it tells you how many feet away from the edge of your house you need to concrete. If you remember from earlier in my letter, we had an equation with all of these three variables in it:

If we plug in the values that we know for X (X = 4.5 feet) and for H (H = 11 feet), we can solve for the last unknown value, B. After plugging in the known values, we find that the value for B = 10 feet. So, as I told you in the very beginning of the letter, you will need to lay the concrete out 10 feet from your house and the width of the concrete will be 12 feet (for the width of glass which is laying on the ground). When this is all constructed, you should expect about 120 squared feet of floor area, and you will be able to walk, without hitting your head, 4.5 feet out from the side of your house towards the glass, and then of course, you will be able to walk 12 feet from one side of the greenhouse to the other (because of width of the glass which sits on the ground is 12 feet). I hope that this explanation helps you to understand where I got the values for the dimensions of your greenhouse. Just a quick recap, here is a picture for you with all the values written in so you know what measurements go to which variables:

Once again, I hope I was of help to you! I wish you much luck with the construction of this greenhouse and I hope that this will get you some business so you can construct for other people in the future!

Your friend,

Sarah Hammel

P. S. I have some people that I would like to thank for helping me to solve this greenhouse problem. I would like to thank Prof. Crannell for letting me come to her office and for letting me ask her questions after class about the solving of the problem (next time I will try to keep track of all "negative" signs!!! Also to Joe and Kyle next door who looked at my derivative in amazement, and then gave advice of how I could solve it. Next to Kelly, A. J., and Pete (in my calc class) with whom I compared my equations, to make sure I was on the right track to solving the problem. And finally to my dad, who avoids math at all causes, who helped me to understand how to explain something to someone who knows nothing about calculus, I pretended I was writing this letter to him.