Quadratic formula ?

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2 Answers
Mar 14, 2018

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.

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Hopefully this helps!

Mar 14, 2018

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.