Question

applications of quadratic relations
date: oct 10/25
instructions: round answers to 1 decimal place if necessary. use proper units of measure. write a concluding statement.

1. nolan has started a custom joke t - shirt business where his profit of p dollars when he sells x t - shirts is represented by the quadratic equation p=-10(x - 50)^2+9000.

a) what is the vertex?
b) what is his maximum profit?
b) how many t - shirts does he have to sell to reach his maximum profit?

1. a ball is thrown off a bridge. the quadratic equation h=-5(t - 6)(t + 4) is a model that gives the approximate height of the ball above the bridge in h metres, at t seconds after it is thrown.

a) what are the zeros of the relation?
b) what do the zeros mean? which one makes sense?
c) what is the axis of symmetry and what does it represent?
d) use the axis of symmetry to find the maximum height of the ball.

Explanation:

Step1: Recall vertex - form of quadratic equation

The vertex - form of a quadratic equation is \(y = a(x - h)^2+k\), and the vertex is \((h,k)\). For the profit equation \(P=-10(x - 50)^2+9000\), \(h = 50\) and \(k = 9000\).

Step2: Find the vertex

The vertex of the quadratic function \(P=-10(x - 50)^2+9000\) is \((50,9000)\).

Step3: Determine maximum profit

Since the coefficient of \((x - 50)^2\) is negative (\(a=-10<0\)), the parabola opens downwards and the \(y\) - coordinate of the vertex is the maximum value. So the maximum profit is \(P = 9000\) dollars.

Step4: Find number of T - shirts for maximum profit

The \(x\) - coordinate of the vertex gives the number of T - shirts to sell for maximum profit. So he has to sell \(x = 50\) T - shirts to reach the maximum profit.

Step5: Find zeros of \(h=-5(t - 6)(t + 4)\)

Set \(h = 0\), then \(-5(t - 6)(t + 4)=0\). Using the zero - product property, \(t-6=0\) or \(t + 4=0\). Solving for \(t\), we get \(t=6\) or \(t=-4\).

Step6: Interpret zeros

The zeros of the function represent the times when the height of the ball above the bridge is \(0\) metres. The zero \(t=-4\) represents a time before the ball is thrown, so it does not make sense in the context of the problem. The zero \(t = 6\) seconds represents the time when the ball hits the water (or the ground level relative to the bridge).

Step7: Find axis of symmetry

The formula for the axis of symmetry of a quadratic function in factored form \(y=a(x - r_1)(x - r_2)\) is \(x=\frac{r_1 + r_2}{2}\). For \(h=-5(t - 6)(t + 4)\), \(r_1=6\) and \(r_2=-4\). So the axis of symmetry is \(t=\frac{6+( - 4)}{2}=\frac{2}{2}=1\) second. It represents the time at which the ball reaches its maximum height above the bridge.

Step8: Find maximum height

Substitute \(t = 1\) into \(h=-5(t - 6)(t + 4)\). \(h=-5(1 - 6)(1 + 4)=-5\times(-5)\times5 = 125\) metres.

Answer:

1.
a. The vertex is \((50,9000)\).
b. The maximum profit is \(9000\) dollars.
c. He has to sell \(50\) T - shirts to reach the maximum profit.
2.
a. The zeros are \(t = 6\) and \(t=-4\).
b. The zeros represent the times when the height of the ball above the bridge is \(0\) metres. The zero \(t = 6\) seconds makes sense as it represents the time when the ball hits the water (or ground - level relative to the bridge).
c. The axis of symmetry is \(t = 1\) second. It represents the time at which the ball reaches its maximum height above the bridge.
d. The maximum height of the ball is \(125\) metres.