NASA launches a rocket at t=0 seconds. Its height, in meters above sea-level, as a function of time is given by h(t)=−4.9t2+169t+178.

Question

NASA launches a rocket at  seconds. Its height, in meters above sea-level, as a function of time is given by . 

Assuming that the rocket will splash down into the ocean, at what time does splashdown occur? 

The rocket splashes down after ___(A)___  seconds. 

How high above sea-level does the rocket get at its peak? 

The rocket peaks at __(B)____  meters above sea-level.

PLEASE ANSWER A AND B!! THANK YOU!!!

Answer 1

A. Rocket splashes down after 35.5127 seconds

B. Rocket peaks at 1635.1939 ft

The numbers above are rounded

Your function is in the form a of quadratic

h(t) = -4.9t² + 169t + 178

h is the height or y component

t is the horizontal or x component

You can write your function in terms of x and y if it helps

y = -4.9x² + 169x + 178

y = ax² + bx + c

when y = 0

0 = ax² + bx + c

0 = -4.9x² + 169x + 178

It can be factored using the Quadratic Formula

The coordinates for the vertex can obtained using -b/2a this gives the t or x coordinate

Then you plug this into to the equation to get h or y coordinate

For your function

h(t) = -4.9t² + 169t + 178

at

h(0)

0 = -4.9t² + 169t + 178

a = -4.9

b = 169

c = 178

(-b±SQRT(b² - 4ac))/2a

(-169±SQRT(169² - 4(178)(-4.9))/2(-4.9)

(-169±SQRT(28561 + 3488.8))/-9.8

(-169±SQRT(32049.8))/-9.8

-169±(179.0246)/-9.8

(-169 + 179.0246)/-9.8 = -1.0229 = t

(-169 - 179.0246)/-9.8 = 35.5127 = t

t cannot be negative

t = 35.5127 seconds

t = -1.0229 h = 0

t = 35.5127 h= 0

For the vertex which will give the maximum height for h

the horizontal component t is

-b/2a = -169/-9.8 = 17. 2449

h(17.2449) = -4.9(17.2449)² + 169(17.2449) + 178

h = 1635.1938 ft

You can graph this at DESMOS.com to check the values.

You can plug the values into the equation to many decimal places.

-4.9(-1.022916257)² + 169(-1.022916257) + 178 = 0

-4.9(1.046357669) + 169(1.022916257) + 178 = 0

-5.127152577 + -172.8728474 + 178 = 0

-178 + 178 = 0

I hope you find this useful if you have any questions send me a message.