Question

A rocket is launched with acceleration of "a ft/sec^2". Neglecting the air resistance, its height after t second is given by f(a, t) = 1/2 (a-32)t^2
feet. Fuel used by the rocket is proportional to a^2 t and the limited fuel capacity of the rocket follows the equation a^2 t = 10000. Find the accelration so that the rocket will reach maximum height when the fuel runs out. What is this height?

Answer 1

Well, first we can express the acceleration as a function of time:
\[a=(\frac{10^4}{t})^{\frac{1}{2}}=\frac{100}{\sqrt{t}}\]If we plug this into our equation for height, we get a function of t alone:
\[f(t)=\frac{1}{2}(\frac{100}{\sqrt{t}}-32)t^2=50t^{\frac{3}{2}}-16t^2\]Now we are asked to maximize this height, so we find the max value of this function with respect to t. Once we have this value of t, the acceleration follows immediately from the relation a^2t=10000

Answer 2

good from here?

Answer 3

not so much! i understood till what you did. What do i need to do after that?

Answer 4

You need to use your calculus techniques to find the max value of this height function

Answer 5

yea. Got it. Thanks

Answer 6

That will give you the height. Whatever t is equal to at this height is used to find the acceleration of the rocket.

Answer 7

so do i do t = 10000/a^2

Answer 8

I get t=5.5s (rounded), giving height=161 ft (rounded) and acceleration=42.7 ft/s^2 (rounded)

Answer 9

Once you get t, you use:\[a=\sqrt{ \frac{10000}{t}}\] to get a.

Answer 10

oooops...I used the wrong expression for a

Answer 11

looks like \[a^{2t}=10000\]I was using\[at^2=10000\]

Answer 12

no no.. its a^2 * t = 10000

Answer 13

so a^2 = 10000/t and then a = sqrt( 10000/t)

Answer 14

eeeer....ok. nm...I did it correct then. I did use \[a^2t=10000\]

Answer 15

:)

Answer 16

any questions?

Answer 17

off to play call of duty...good luck

Answer 18

I am struggling to find t

Answer 19

\[f'(t)=\frac{150}{2}t^\frac{1}{2}-32t=0\]\[\sqrt{t}= \frac{150}{64}\]So,\[t \approx5.5\]

Answer 20

awesome and i did get a as 42.7 and max height at 161 rounded.
Now as an additional information what do i do to find the value of a that maximizes the total height of the rocket. And then for this, what is the maximum height. How do i proceed with that?

Answer 21

fuel is a limitation. use lagrange multiplier

Answer 22

and can you show me how to do that? please. :(

Answer 23

just wondering. you know what lagrange multiplier is right?

Answer 24

yea its used to fine max/min values.

Answer 25

k then ill show u the first few steps

Answer 26

hold on. will take a while to draw out