2D geometry – Helping out on the ranch (advanced)

How does the bird’s eye view of the barn look geometrically? Try to sketch it. The stalls will have one of their shorter sides touching the path.

What lengths contribute to how long the barn has to be? And to how wide it is to be?

How much is there to paint? How big are each of the barn’s walls?

Knowing how much there is to paint, how can you get the cost to paint it?

If the area you are supposed to paint corresponds to a set amount of money, how can we calculate the additional 10% we need to buy?

length = 42 ft, width = 26ft

cost = $120

Let’s try to draw what the barn is supposed to look like. On each side, there are supposed to be 7 stalls, one adjoining the other, and each should have length stall_length = 8 ft and width stall_width = 6 ft. Therefore, the length of the barn will be equal to how long the 7 stalls will be when put one next to the other. Remember that we are arranging them to so that they face the path with their narrow sides. In other words,
length = 7 * stall_width = 7 * 6 ft = 42 ft.

To compute the width of the barn, we have to see what are its parts. On one side we have the stalls, in the middle we have the path, and to the right again we have stalls. The stalls are faced with their narrow side (their width) towards the path. This means that they contribute their longer side to the barn’s width. This, together with the 10-foot-wide path, gives
width = stall_length + 10 ft + stall_length = 8 ft + 10 ft + 8 ft = 26 ft.

We begin by calculating the area we need to paint, which is the area of all the walls of the barn. Since the barn is rectangular, each of its four walls is also a rectangle. What is more, all of them are the same height, namely h = 15 ft. Therefore, this height is one of the sides of every rectangle that is a lateral face of the barn (a fancy, mathematical name for a wall). The other side is either the barn’s width (for two of the four walls) or the barn’s length (for the remaining two). Fortunately, we got both of them in question 1.

The area of a rectangle calculator comes to help us and gives
area = 2 * h * width + 2 * h * length = 2 * 15 ft * 26 ft + 2 * 15 ft * 42 ft = 2,040 ft².

NOTE: There is a different way of calculating the area we need to paint. We still need to calculate the area of all the walls of the barn, but why not do all four of them at once? To do this, we first observe that all of them have the same height h = 15 ft. Therefore, each side of the rectangle that is the base of the barn (as drawn in question 1) has to be multiplied by the height h to return the area of the wall. So to calculate the total area of all the four walls, we can start by using the perimeter of a rectangle calculator to obtain the barn perimeter:
perimeter = 2 * width + 2 * length = 2 * 26 ft + 2 * 42 ft = 136 ft.
And now we finish by multiplying this by the height h:
area = perimeter * h = 136 ft * 15 ft = 2,040 ft².

Now we move on to the paint. The scenario suggests that every gallon can cover 400 square feet. Therefore, we will need
area / 400 = 2,040 / 400 = 5.1 US gal
of paint to do the whole barn. However, we need to add the extra paint for the potential spillage accidents. The scenario tells us that we should add 10% of what we got above. This means that we need 100% + 10% = 110% of that number. The percentage calculator gives that this is
110% * 5.1 US gal = 5.61 US gal.

Since we’re buying paint in half-gallon cans, in total we’ll need
5.61 / 0.5 = 11.22
cans. Clearly, we can’t buy 0.22 of a can, so in the end we must buy 12 of them. The scenario tells us that one costs $10, so all of them will cost
cost = 12 * $10 = $120.