Expert: Paul Klarreich - 6/6/2006

Question



Question -
hi, please help me with this difficult, difficult, difficult question!!

Suppose a bullet is fired on a distant planet so that its height (in feet) after t seconds is given by h= -4t^2+16t+5.
A. when is the bullet at its highest point?
B. when will the bullet strike the ground?
c. what is the maximum height the bullet will attain?


thank you soooooooooooooooooooo much for your time! :)
please show me how you did it.

Answer
Hi, Hannah,
You wrote:
Subject:algebra, quadratic models
Question -
hi, please help me with this difficult, difficult, difficult question!

Suppose a bullet is fired on a distant planet so that its height (in feet) after t seconds is given by h= -4t^2+16t+5.
A. when is the bullet at its highest point?
B. when will the bullet strike the ground?
c. what is the maximum height the bullet will attain?
--------------------------------------

OK, since I conclude that you are not studying calculus, you must handle it this way:

The graph of  h = -4t^2 + 16t + 5  using h as the vertical axis (logical, of course) is a parabola with its vertex at the top.  Analysis of the parabola gives the following:

For any parabola of the form  h = ax^2 + bx + c

A. the vertex is at  t = -b/2a = -16/(-8) = 2.
At t = 2, h = -4(2)^2 + 16(2) + 5 = -4(4) + 32 + 5 = 21 feet.

So the bullet reaches maximum height at t = 2, and that maximum is 21 feet.

B. It will strike the ground when h = 0.  Solve:

-4t^2 + 16t + 5 = 0

4t^2 - 16t - 5 = 0

4(t^2 - 4t + 4) = 5 + 4(4)
4(t - 2)^2 = 21
t - 2 = +- sqrt(21)/2

t = 2 +- sqrt(21)/2

The negative square root gives us a negative number, which we discard, so the solution is

t = 2 + sqrt(21)/2 , approximately   4.3 seconds.