Equations and Inequalities
If you don't remember them, it is a good idea to review your geometry formulas. They will come back to haunt you all the time, and it is better if they are more like Casper the Friendly Ghost than Jasper the Grumpy Ghost Who Will Scare the Bejeezus out of You in the Middle of the Night. So strange that that Saturday morning cartoon never got off the ground.
If a triangle has height 4 cm and area 20 cm2, how long is the base of the triangle?
This is a fairly straightforward question, as long as you've memorized your geometry formulas. Ahem. The area A of a triangle is given by , so plug in the numbers given in the problem to find that
We are multiplying the fraction, the variable, and the 4 together. The variable we can't do anything about for the time being, but we can multiply the fraction and the 4 together to get 2. We can then divide both sides by 2 to find that b = 10 cm.
In the picture below, find a formula for a2 in terms of the shaded area. Let A denote the shaded area.
The shaded area A is equal to the area of the square minus the area of the circle. There's no special formula for that part—you can figure out that part by eyeballing it. Or by smelling it, or whatever your most reliable sense is. The side length of the square is a, so the area of the square is a2.
The radius of the circle is , and the area of the circle is
Putting together all the puzzle pieces, we conclude that
We're almost done. The only thing left is to actually do what the question asks, which is to provide a formula for a2 in terms of A. With whipped cream and a cherry on top, if possible. We need to rearrange our current formula a bit. First, simplify the right-hand side to find that
There are two ways we can go from here. Technically, there are infinitely many ways we can go from here, but only two correct ones. Since time is limited, we will only go over the correct ones.
Way 1: Factor out the a2 from each term in the right-hand side to find that
Divide each side by the quantity in parentheses, and here we are:
Way 2: Instead of factoring out the a2 right away, multiply both sides of the simplified formula for A by 4 to get rid of fractions. Then, we find that
4A = 4a2 – π a2
which is much prettier. In fact, we would kiss it on the mouth if it had one.
Now factor out a2 to find that
4A = a2(4 – π)
and divide by the quantity in parentheses to find that