Building a Greenhouse

Question

Building a Greenhouse
Your parents are going to knock out the bottom of the entire length of the south wall of their house and turn it into a greenhouse by replacing the bottom portion of the wall with ahuge sloped piece of glass (which is expensive). They have already decided they are going tospend a certain ?xed amount. The triangular ends of the greenhouse will be made of variousmaterials they already have lying around.

The ?oor space in the greenhouse is only considered usable if they can both stand up init, so part of it will be unusable. They want to choose the dimensions of the greenhouse to getthe most usable ?oor space. What should the dimensions of the greenhouse be and how muchusable space will your parents get?

Answer 1

To maximize the usable floor space in the greenhouse, your parents should aim to create a rectangular shape with the glass sloped roof. The dimensions should be chosen so that they both can stand comfortably in it while using the least amount of expensive glass for the sloped portion.

Here's how you can approach this problem:

1. **Determine the Maximum Height:** First, consider the maximum height your parents can comfortably stand. Let's say this height is "H" meters.

2. **Create a Rectangle:** The sloped glass roof will create a triangular shape on each side of the rectangular base. The width of the rectangle can be set as the width of the greenhouse (W).

3. **Determine the Length of the Rectangular Base:** To maximize the usable space, the length of the rectangular base (L) should be chosen such that the triangle's hypotenuse (the sloped glass) is equal to H.

4. **Calculate the Usable Floor Space:** The usable floor space is the area of the rectangular base.

Now, let's calculate the usable floor space:

The area of a rectangle is given by:

\[Area_{\text{rectangle}} = \text{Length} \times \text{Width} = L \times W\]

The area of a right triangle is given by:

\[Area_{\text{triangle}} = \frac{1}{2} \times \text{Base} \times \text{Height} = \frac{1}{2} \times W \times H\]

Since there are two identical triangular ends, the total area of the triangular ends is:

\[2 \times \frac{1}{2} \times W \times H = W \times H\]

Now, the usable floor space in the greenhouse (UsableArea) is:

\[UsableArea = Area_{\text{rectangle}} - 2 \times Area_{\text{triangle}}\]

\[UsableArea = (L \times W) - (W \times H)\]

To maximize this area, you want to differentiate with respect to L (the length of the rectangular base), set the derivative equal to zero, and solve for L. This will give you the value of L that maximizes the usable floor space.

Once you find the value of L, plug it back into the expression for usable area to find the maximum usable floor space.

By following this approach, you can determine the dimensions of the greenhouse that will give your parents the most usable floor space within their budget and the maximum height they can comfortably stand in.