Parametric Parabolas

It hasn't even been a week since we dealt with parabolas, and they come again, this time with actual numbers. The question itself isn't hard, but once again, some thinking is required.

A parabola goes through the points A(17, 3) B(34, z) C(50, -30). B is the maximum point of the parabola. Find point B and the parabola function in standard form.

As you know, a parabola can be written in two ways. One is standard form, with the a, b, and c coefficients, and one is the vertex form. To solve this question, we'll use the vertex form. This form says that the equation a parabola that has a vertex (h, k) is:
y = a(x - h)² + k

We know h and k, so let's plug them in:
y = a(x - 34)² + z

We have two other points, so let's use them:
3 = a(17 - 34)² + z
2 = a(50 - 34)² + z

3 = 289a + z
-30 = 256a + z

Subtract the second equation from the first:
33 = 33a
a = 1
3 = 289a + z
3 = 289 + z
z = -286

The function of the parabola is:
y = (x - 34)² - 286

Open parenthese and combine like terms to switch to standard form:
y = x² - 68x + 1156 - 286
y = x² - 68x + 870

Nice, isn't it?
Nadav

nadavs

Lawn Mowing Siblings

After yesterday's easy question I found something better. It doesn't deal with differential equations or have a fancy name, but it just deals with some thinking and setting up equations correctly.

A brother and a sister are mowing the lawn on their backyard. It takes the brother 15 minutes longer to mow the lawn than it takes his sister. Together they mow the lawn in 56 minutes. How long does it take each of them to mow the lawn?

To solve this, we need something, and we need it fast: variables. x sounds like a great variable. Let x be the time it takes the sister to mow the lawn.

Now, since it takes her x minutes to mow the lawn, she does 1/x of the lawn every minute. Her brother takes x + 15 minutes to mow the lawn, so he mows 1/(x + 15) every minute.

Since together they mow the lawn in 56 minutes, 56 times the mowing rate of the brother plus 56 times the mowing rate of the sister should give 1, which is 100% of the lawn. We finally have an equation:
56/x + 56/(x + 15) = 1

This equation is easily solved:
56(x + 15) + 56x = x(x + 15)
56x + 840 + 56x = x² + 15x
x² - 97x - 840 = 0

Now we can solve it by factoring or by the quadratic formula. Since the quadratic formula is boring, let's try to factor. The numbers here are big, so you can start from something like 10 and see how the difference between the factors grow. Finally, you will reach the correct factoring:
(x - 105)(x + 8) = 0

Since the sister can't mow the lawn in -8 minutes, it must be 105 minutes, or 1 hour and 45 minutes. This means the brother needs 120 minutes, or two whole hours to mow this lawn. It must be a big lawn.

To test that, check: 56/105 + 56/120 = 8/15 + 7/15 = 1

I really doubt this answer is correct. If it takes the sister alone almost two hours to mow the lawn, with her brother it will take them about 3-4 hours, and that's before they even start mowing.

There is a new proof added at super math tips, go check it out.

Have a great weekend,
Nadav

nadavs