# Hard GCSE Question

(5) Sue is making a toy rocket in her science lesson which is to be launched from the
ground.
The flight path of the toy rocket can be modelled by the equation 2
h 2t t 6 1. h is the height in metres the rocket reaches above the ground.
t is the time in seconds after the rocket is launched.
Find the maximum height above the ground that the rocket reaches and the time it
takes to reach this height.
And with symbols:
Original post by Reyaansh
(5) Sue is making a toy rocket in her science lesson which is to be launched from the
ground.
The flight path of the toy rocket can be modelled by the equation 2
h 2t t 6 1. h is the height in metres the rocket reaches above the ground.
t is the time in seconds after the rocket is launched.
Find the maximum height above the ground that the rocket reaches and the time it
takes to reach this height.

Post where you've got to
For a head start, there is usually one (reasonable) thing to try when you see a quadratic and the word "maximum".
So far I factorised it down to h = -2([2-1.5]^2-2.75 but am extremely confused on where to go from here
Original post by Reyaansh
So far I factorised it down to h = -2([2-1.5]^2-2.75 but am extremely confused on where to go from here

Okay, you know you should employ the "completing square" technique - that's a very good idea.

Also something that's alarming at first sight is somehow the t's are gone. That can't be right.

Now some tips:
- You can check your algebra is correct if you re-expand your answer. You should get what you started with.
- Do your algebra s l o w l y but accurately. I guess that would be the appropriate kind of emphasis?

Sidenote: There is another big tell that your algebra must be incorrect with respect to this problem itself, if you know what your goal for completing square is. i.e. What is the connection between completing square and maximum of a quadratic?
(edited 10 months ago)
Oh no ive forgotten the T's 😭😭. A really silly mistake. But this is a really good hint. thank you so much, I'll try it again
Original post by Reyaansh
(5) Sue is making a toy rocket in her science lesson which is to be launched from the
ground.
The flight path of the toy rocket can be modelled by the equation 2
h 2t t 6 1. h is the height in metres the rocket reaches above the ground.
t is the time in seconds after the rocket is launched.
Find the maximum height above the ground that the rocket reaches and the time it
takes to reach this height.

Firstly in order to find if the turning point of the function is a maxima or a minima, we must find the second order derivative of the function. If it is less than 0, the turning point is a maxima and it is more than 0, the turning point is a minima. So, let h=-2t^2 +6t +1. dh/dt = -4t+6, ((d^2)h))/(d(t^2))= -4 which is less than 0 and the turning point is the maxima, the highest height of the function.
Then in order to find the turning point, we can use the first order derivative and at the turning point, dh/dt = 0. So, let -4t+6=0, subtract 6 to get -4t=-6 , then divide by -4 to get t = -6/-4 = 3/2 = 1.5 seconds to reach the maximum height. Then we substitute the 3/2 seconds back in to the function to get -2(9/4) + 6(3/2) + 1 = -9/2 + 9 + 1 = 10-(9/2) = (20-9)/2 = 11/2 = 5.5 metres which is the maximum height above the ground and it takes 1.5 seconds to reach this height above the ground.
Original post by π/2=Σk!/(2k+1)!!
Firstly in order to find if the turning point of the function is a maxima or a minima,

Please don't post full solutions - it's against the rules of the forum!

Also, just as a point of terminology, note that "maxima" and "minima" are plurals - you want to find out whether a turning point is "a maximum" or "a minimum"