Question

Quadratic formula ?

Answer 1

Alex will land `3.9` meters from the ramp.

Explanation:

We seek to find `h(d) = 0`.

`0 = -3.9d^2 + 13.1d + 8.7`

Now apply the quadratic formula.

`d = (-13.1 +- sqrt(13.1^2 - 4 * -3.9 * 8.7))/(2 * -3.9)`

`d = (-13.1 +- sqrt(307.33))/(-7.8)`

We now use a calculator.

`d = 3.927 m or -0.568` m

A negative answer is clearly impossible so Alex will land `3.9` meters away from the ramp. The graph of the `h(d)` function confirms.

Hopefully this helps!

Answer 2

`19.7` metres in height.

Explanation:

It's stated in this problem that Alex's path can be modelled by the quadratic function `h(d) = - 3.9d^(2) + 13.1d + 8.7`.

The maximum value of this function will be the maximum height reached by Alex.

In order to find the maximum value of `h(d)`, we need to complete the square:

`Rightarrow h(d) = - 3.9d^(2) + 13.1d + 8.7`

`Rightarrow h(d) = - 3.9 (d^(2) + frac(13.1)(- 3.9)d + frac(8.7)(- 3.9))`

`Rightarrow h(d) = - 3.9 (d^(2) - frac(13.1)(3.9)d - frac(8.7)(3.9))`

`Rightarrow h(d) = - 3.9 (d^(2) - frac(13.1)(3.9)d + (frac(frac(13.1)(3.9))(2))^(2) - frac(8.7)(3.9) - (frac(frac(13.1)(3.9))(2))^(2))`

`Rightarrow h(d) = - 3.9 (d^(2) - frac(13.1)(3.9)d + (frac(13.1)(7.8))^(2) - frac(8.7)(3.9) - (frac(13.1)(7.8))^(2))`

`Rightarrow h(d) = - 3.9 ((d- frac(171.61)(60.84))^(2) -5.0514464169)`

`Rightarrow h(d) = - 3.9 (d- frac(171.61)(60.84))^(2) + 19.70064103`

Now, the expression `(d- frac(171.61)(60.84))^(2)` is always positive, as it is squared, i..e it is always greater than or equal to zero.

`Rightarrow (d- frac(171.61)(60.84))^(2) geq 0`

Multiplying this expression by `- 3.9` (a negative number) reverses the inequality:

`Rightarrow - 3.9 (d- frac(171.61)(60.84))^(2) leq 0`

Let's add `19.70064103` to both sides of the inequality:

`Rightarrow - 3.9 (d- frac(171.61)(60.84))^(2) leq 19.70064103`

This expression is now equivalent to our function `h(d)` (in vertex form):

`therefore h(d) leq 19.70064103`

According to this inequality, the function `h(d)` is less than or equal to `19.70064103`, i.e. has a maximum value of `19.70064103`.

Rounding this number to the nearest tenth gives `h(d) leq 19.7`.

Therefore, Alex reaches no higher than `19.7` metres.