Question

A cliff diver dives from 17m above the water. The diver’s height above the water, h(t) in metres after t seconds is modelled by h(t) = -4.9t2 + 1.5t + 17. Determine when the diver was 5 m above the water.
Please help :(

Answer 1

Answer:

If you want to round to the nearest hundredths, the answer is 1.73 seconds.

Step-by-step explanation:

So we want to solve h(t)=5 for t because this will give us the time,t, that the diver was 5 m above the water.

$-4.9t^2+1.5t+17=5$

My goal here in solving this equation is to get it into $at^2+bt+c=0$ so I can use the quadratic formula to solve it.

The quadratic formula is $t=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$.

So let's begin that process here:

$-4.9t^2+1.5t+17=5$

Subtract 5 on both sides:

$-4.9t^2+1.5t+12=0$

So let's compare the following equations:

$-4.9t^2+1.5t+12=0$

$at^2+bt+c=0$.

$a=-4.9$

$b=1.5$

$c=12$

Now we are ready to insert in the quadratic formula:

$t=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$t=\frac{-1.5 \pm \sqrt{(1.5)^2-4(-4.9)(12)}}{2(-4.9)}$

I know this can look daunting when putting into a calculator.

But this is the process I used on those little calculators back in the day:

Put the thing inside the square root into your calculator first.  I'm talking about the $(1.5)^2-4(-4.9)(12)$.

This gives you:  237.45

Let's show what we have so far now:

$t=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$

$t=\frac{-1.5 \pm \sqrt{(1.5)^2-4(-4.9)(12)}}{2(-4.9)}$

$t=\frac{-1.5 \pm \sqrt{237.45}}{2(-4.9)}$

I'm going to put the denominator, 2(-4.9), into my calculator now.

$t=\frac{-1.5 \pm \sqrt{237.45}}{-9.8}$

So this gives us two numbers to compute:

$t=\frac{-1.5 - \sqrt{237.45}}{-9.8} \text{ and } t=\frac{-1.5+\sqrt{237.45}}{-9.8}$

I'm actually going to type in -1.5-sqrt(237.45) into my calculator, as well as, -1.5+sqrt(237.45).

$t=\frac{-16.90941271}{-9.8} \text{ and } t=\frac{13.90941271}{-9.8}$

We are going to use the positive number only for our solution.

So we have the answer is whatever that first fraction is approximately:

$t=\frac{-16.90941271}{-9.8}=1.725450277$.

The answer is approximately 1.73 seconds.