GKT101: General Knowledge for Teachers – Math

1. $15$ meters

The height of the stone at the time it is thrown is given by $h(0)$.

$\begin{aligned} h(0) &=-5(0)^{2}+10(0)+15 \\ &=0+0+15 \\ &=15 \end{aligned}$

In conclusion, the height of the stone at the time of throwing is $15$ meters.

2.

The garden's area is modeled by a quadratic function, whose graph is a parabola.

The maximum area is reached at the vertex.

So in order to find when that happens, we need to find the vertex's $w$-coordinate.

The vertex's $x$-coordinate is the average of the two zeros, so let's find those first.

$\begin{gathered} A(x)=0 \\ -x^{2}+100 x=0 \\ x^{2}-100 x=0 \\ x(x-100)=0 \\ \swarrow \searrow\\ x=0 \text { or } x-100=0 \\ x=0 \text { or } x=100 \end{gathered}$

Now let's take the zeros' average:

$\frac{(0)+(100)}{2}=\frac{100}{2}=50$

In conclusion, the maximum garden area occurs when the width is $50$ meters.

3. $8$ thousand dollars.

The company's profit is modeled by a quadratic function, whose graph is a parabola.

The maximum profit is reached at the vertex.

So in order to find the maximum profit, we need to find the vertex's $y$-coordinate.

We will start by finding the vertex's $x$-coordinate, and then plug that into $P(x)$.

The vertex's $x$-coordinate is the average of the two zeros, so let's find those first.

$\begin{gathered} P(x) =0 \\ -2 x^{2}+16 x-24 =0 \\ x^{2}-8 x+12 =0 \\ (x-2)(x-6) =0\\ \swarrow \searrow \\ x-2=0 \text { or } x-6=0 \\ x=2 \text { or } x=6 \end{gathered}$

Now let's take the zeros' average:

$\frac{(2)+(6)}{2}=\frac{8}{2}=4$

The vertex's $x$-coordinate is $4$. Now let's find $P(4)$:

$\begin{aligned} P(4) &=-2(4)^{2}+16(4)-24 \\ &=-32+64-24 \\ &=8 \end{aligned}$

In conclusion, the maximum profit is $8$ thousand dollars.

4. $4$ seconds

The ball hits the ground when $h(x)=0$.

$\begin{gathered} h(x)=0 \\ -2 x^{2}+4 x+16=0 \\ x^{2}-2 x-8=0 \\ (x+2)(x-4)=0 \\ \swarrow \searrow \\ x+2=0 \text { or } x-4=0 \\ x=-2 \text { or } x=4 \end{gathered}$

We found that $h(x)=0$ for $x=-2$ or $x=4$. Since $x=-2$ doesn't make sense in our context, the only reasonable answer is $x=4$.

In conclusion, the ball will hit the ground after $4$ seconds.