The zeros: when the ball lands

The stone of the last lesson was thrown from a cliff, and its height above the water after $\displaystyle t$ seconds is $\displaystyle {h}={-5t^{2}+20t+25}$.
It hits the water when the height is 0, so we solve the equation:

• We divide both sides by $\displaystyle -5$: $\displaystyle {t^{2}-4t-5}={0}$.
• By the quadratic formula, with $\displaystyle {a}={1}$, $\displaystyle {b}={-4}$, $\displaystyle {c}={-5}$: $\displaystyle {t_{1,2}}={\frac{4\pm\sqrt{16+20}}{2}}={\frac{4\pm6}{2}}$.
• $\displaystyle {t}={5}$ or $\displaystyle {t}={-1}$.

A negative time has no meaning, so the stone hits the water after 5 seconds.

When is the stone 40 metres high? This time the height is 40, not 0:

• We take 40 over to the left side: $\displaystyle {-5t^{2}+20t-15}={0}$, and divide by $\displaystyle -5$: $\displaystyle {t^{2}-4t+3}={0}$.
• $\displaystyle {t}={1}$ or $\displaystyle {t}={3}$.

On the graph, these are the points where the parabola meets the line $\displaystyle {y}={40}$:

• After one second the stone passes the height of 40 on the way up, and after 3 seconds on the way down. Both solutions are right.
• The line $\displaystyle {y}={50}$ does not meet the parabola at all: the equation $\displaystyle {-5t^{2}+20t+25}={50}$ has no solution, because the stone’s greatest height is 45 metres.

Sometimes the root is not a whole number. A fountain’s jet comes out one metre high, and its height $\displaystyle x$ metres away is $\displaystyle {y}={-x^{2}+4x+1}$. Where does the water hit the ground?

• $\displaystyle {-x^{2}+4x+1}={0}$, and by the quadratic formula: $\displaystyle {x}={2+\sqrt{5}}$ or $\displaystyle {x}={2-\sqrt{5}}$.
• $\displaystyle 2-\sqrt{5}$ is negative – behind the pipe, and has no meaning.
• $\displaystyle 2+\sqrt{5}\approx4.24$.

The water hits the ground about 4.24 metres from the pipe.

A boy throws a ball from a balcony 20 metres high.
The ball’s height above the yard, in metres, $\displaystyle t$ seconds after the throw: $\displaystyle {h}={-5t^{2}+15t+20}$

Solve the equation. After how many seconds does the ball hit the yard?

When is the ball 30 metres high?

Does the ball reach a height of 35 metres? Solve, and explain the answer.

A sprinkler one metre above the lawn sprays water.
The water’s height, in metres, $\displaystyle x$ metres from the sprinkler: $\displaystyle {y}={-0.5x^{2}+2x+1}$

Solve the equation.

How far from the sprinkler does the water hit the lawn? Round to two decimal places.

The number of people ill with flu in a town, in hundreds, $\displaystyle x$ weeks after the start of winter: $\displaystyle {y}={-x^{2}+10x+24}$
Mark on the board the points where 4,000 people are ill, and check.
In which weeks were 4,000 people in the town ill? When was the flu at its peak, and how many were ill then? When, by the model, does the flu disappear?