Question

The height of a rocket, h, is increasing at a constant rate of 18 feet per second.

Answer 1

Answer:

Step-by-step explanation:

I'd use calculus to address this problem.  Not knowing whether or not you're in calculus, I'm introducing the standard equation for vertical motion subject to gravity:

h(t) = h(0) + v(0)*t - 9.8 t^2

If its height at 5 seconds is 118 feet, then write an equation for h as a function of time, t, in seconds since it was fired.  That would be:

h(5) =  h(0) + (18 ft/sec)(5 sec) - (1/2)*(32 ft/sec^2)(5 sec)^2 = 118 ft  

Our job here is to determine the initial height, h(0).  Performing the indicated mult., we get:

h(0) + 90 ft - 400 ft = 118 ft

This results in h(0) = 118 ft - 90 ft + 400 ft, or 428 ft.

Question 1:  

The desired equation is h(t) = 428 ft + (18 ft/sec)*t - (16 ft/sec^2)*t^2.

Question 2:

Assuming that this equation is correct, we can set it = to 190 ft and solve for the time, t:

190 ft = 428 ft + (18 ft/sec)*t - (16 ft/sec^2)*t^2.  We need to solve this quadratic equation for t:

-16t^2 + 18t + 428 - 190 = 0, or -16t^2 + 18t + 238 = 0

Determining the coefficients of the t powers:  a = -16, b = 18 and c = 238

Then the determinant is b^2-4(a)(c), which here is:

18^2-4(-16)(238) = 15556, and the square root of that is 124.7.

Thus, the two possible times are:

      -18 plus or minus 124.7            -18 plus or minus 124.7

t = --------------------------------------- = ---------------------------------

                 2(-16)                                             -32

Thus, t = -3.34 sec and t = 4.46 sec

The rocket will be 190 ft above the ground after 4.46 sec.

Question 3:  Just take that '428 ft' from the equation found in Question 2:  428 ft. above the ground when launched

Answer 2

1)  Equation for height, h(t) =  (18t + 28) ft at time t seconds.

2)After 9 seconds height of rocket is 190 ft

3) Initial height of rocket is 28 ft

Step-by-step explanation:

Rate of increasing of height = 18 ft/s

1) Given that at t = 5 seconds height is 118 feet,

       That is

                        h(5) = 118 ft

        We know that height of rocket is increasing 18 ft in each second, and it will also have an initial height.

        So

                    h(t) = 18t + h₀

                    118 = 18 x 5 + h₀

                     h₀ = 118 -90 = 28 ft

        Equation for height, h(t) =  (18t + 28) ft at time t seconds.

2) Now we need to find time when height of rocket is 190 ft

             h(t) =  (18t + 28) = 190

              18t = 190-28 = 162

                 t = 9 seconds

      After 9 seconds height of rocket is 190 ft

3) Here we need to find initial height of rocket, that is t = 0 seconds

               h(0) =  (18x0 + 28) = 28 ft

              Initial height of rocket is 28 ft