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

Maximizing School Profit

Today's question is dealing with schools. As you know, schools need money, so they send dance tickets. However, they need to know how much to charge. Here is a question dealing exactly with that.

A school is holding a dance with a ticket price of $6. 250 plan to attend. The dance team knows that for every $1 increase in the ticket price, 25 less people will come. How much should a ticket cost for maximum revnue? Solve using two different methods.

First, we need an equation to solve, otherwise we won't be able to solve it even with one method. As we know, for each $1 price increase, 25 less people will come. So let x be the number of $1 price increases.

When you increase the price x times, the price goes up by $x ($1 * x). When the price is increased x times, 25x less people will come. The profit is the number of people times the price per ticket. Since the starting price is 6, one factor will be (6 + x). Since 250 plan to attend initially, the second factor is (250 - 25x). This makes the entire profit function:
P = (x + 6)(250 - 25x)
P = -25x² + 100x + 1500

To solve that, we can use two different methods. The first one is using vertex form, which will give us the x coordinate of the vertex, and reveal the maximum point that way (it's an upside-down parabola).
P = -25x² + 100x + 1500
P = -25(x² - 4x - 60)

Complete the square inside:
P = -25(x² - 4x + 4 - 64)
P = -25(x - 2)² + 1600

This means the maximum profit is generated when x = 2 and the ticket price is $8. In this case, 200 people will come, and the profit will be $8 * (250 - 25*2) = $8 * 200 = $1600.

To solve the other way, differentiate P:
P' = -50x + 100

Compare to zero:
0 = -50x + 100
50x = 100
x = 2

Once again, the same result.

Hope you liked it,
Nadav

nadavs