Trig and algebra 2 help?!?! -16x^2+70x+37?

Question

Ok i have to do a sheet for trig and dont know what to do wit this problem:

The height (Y) of maria's rocket can be represented by Y= -16x^2+70x+37 where x represents the time in sesconds since the rocket was launched, and the height is expressed in feet. What was teh maximum height of maria's rocket to the nearest tenth of a foot

I would like someone to explain to me what to do please and thanks,
Allie B.

Answer 1

First notice that the equation is a concave down parabola (since it is a polynomial of degree 2 with first coefficient being negative).

Therefore, finding the vertex will suffice finding the maximum. To do this, you must complete the square as follows:

Y/-16 = x^2 - (70/16) x - 37/16

- Y/16 = x^2 - (70/16)x +(35/16)^2 - (35/16)^2 - (37/16)

- Y/16 = (x - 35/16)^2 - (35/16)^2 - (37/16)

Y = -16 (x- 35/16)^2 + (35^2/16) + 37

So Maximum is at:

x = 35/16 = 2.1875 seconds
Y= (35^2/16) + 37 = 113.5625 ft.

Answer 2

f (x) = - 16x ² + 70x + 37
f `(x) = - 32 x + 70 = 0 for max. value
32 x = 70
x = 70 / 32 = 35 / 16
Max. height
= (-16) (35/16) ² + 70(35/16) + 37
= - (35 ²) / 16 + 2450 / 16 + 37
= - (1225) / 16 + 2450 / 16 + 37
= 1225 / 16 + 37
= 113.6 ft