Question

How long will it take for the rocket to reach maximum height?

Equation: h= -5t^2+30t+10

Please show me how to do this

Answer 1

Take derivative, set equal to zero, solve for t.

Why does this work?

When you set the derivative equal to zero, you are asking the question, when is the function h flat (i.e. has zero slope)? The function h changes direction and goes from positive slope to negative slope at the peak of the flight path, which is at the apex of the parabola. When something goes from positive to negative, it must go through zero. Hence, setting derivative of h to zero is asking when is the slope zero - this locates the apex of h.

Make sense?

http://www.wolframalpha.com/input/?i=d%2Fdt+%28+-5t%5E2%2B30t%2B10%29+%3D+0

http://www.wolframalpha.com/input/?i=h%3D+-5t%5E2%2B30t%2B10

Answer 2

An alternative approach, perhaps shorter but not as exciting mathematically, would be to find the vertex of the given quadratic function. The first coordinate of this vertex would be the time at which the rocket reaches its max. height.

Answer 3

So finding the time at which the rocket hits the max height is the same way to find the time in which the rocket would land?

Answer 4

@ybarrap @mathmale

Answer 5

@ParthKohli can you please help me

Answer 6

think of `h = -5t^2 + 30t + 10` as `y = -5x^2 + 30x + 10`

Answer 7

`y = -5x^2 + 30x + 10` is in the form `y = ax^2 + bx + c`

can you tell me what 'a', 'b', & 'c' are?

Answer 8

any ideas @ohohaye ?

Answer 9

a=5 b=30 c=10

Answer 10

a = -5 actually

Answer 11

Oops, didn't see the negitive sign

Answer 12

now you need to use this formula

\[\Large h = -\frac{b}{2a}\]

Answer 13

\[\Large h = -\frac{b}{2a}\]


\[\Large h = -\frac{30}{2(-5)}\]


\[\Large h = ???\]

Answer 14

h=-3

Answer 15

the two negatives will cancel to form a positive

Answer 16

Oh ok, so it's just 3?

Answer 17

yes, h = 3 is where the rocket reaches the peak height

Answer 18

Ok, so I would use the \[h=-\frac{ b }{ 2(a) }\] if I had a question like this on a test or something?

Answer 19

yes that will get you the x coordinate of the vertex

Answer 20

I used this to find the max height of the rocket though

Answer 21

in this case, t takes place of x

so if we know the x coordinate of the vertex, then we know how long it takes til the object reaches the peak

Answer 22

to find the max height, you plug t = 3 back into the original equation

Answer 23

Oh, so to find the time, I only have to use the first formula, but the find the max height I have to plug the answer back into the equation?

Answer 24

correct

Answer 25

what happens when you plug in t = 3 ?

Answer 26

I get 55

Answer 27

so 55 is your max height

Answer 28

Yep