word problem solving help please

Question

The altitude of the plane (in meters above the ground) n minutes after takeoff is modelled

h(n) = 45 - 0.3n² + 16n

1. What is the planes greatest altitude?
2. To the nearest minute, how long would it take to reach this altitude?
3. To the nearest minute, when does the helicopter land on the ground?
4. The plane gets hit mid-flight by an object following the path h(n) = 25 + 6n

a) To the nearest second, find & describe the point of impact.

Answer 1

1. h(n) = - 0.3(n² - 160/3n - 150) = - 0.3(n - 80/3)² + 45 + 640/3 = 775/3 - 0.3(n - 80/3)^2;

max h(n) = 775/3 m - 80/3 = 0;

2.n = 80/3 = 27 min

3.h(n) = 0; n = 80/3 ±√7750/9; n = 80/3 + 5√310/3 = 56 min

4.- 0.3n² + 16n + 45 = 25 + 6n; - 0.3n² + 10n + 20 = 0; n = ( 5 ±√25 + 6)/0.3

n = (5 + √31)/0.3 = 35 min

a) h = 25 + ·6·35.23 = 236.4 m

Answer 2

I am not going to answer all the questions, but show you how to approach them:

1) take the derivative of the function, equate it to 0, and solve for n. then plug in that n into f(n) to get the max height.

Algebraically, you can find the vertex of the parabola n = -b/2a for parabola in the form ax²+bx+c =0 (plug into f(n) for height)

2) n in problem 1

3) I assume that you mean the plane reaching f(n)=0. solve f(n) = 0 using quadratic formula.

4) time of impact will be for n such that f(n) = 25 + 6n. Again, solve the quadratic for n and then plug in that n into either equation to find the height of impact.