A dud missile is fired straight into the air. The missile's height is given by the formula: h(t)=-16t+400t+100. At what time will the missile reach its maximum height?

Question

A dud missile is fired straight into the air. The missile's height is given by the formula: h(t)=-16t+400t+100.

At what time will the missile reach its maximum height?

What is the maximum height of the missile?

When will the missile reach 2,500 feet above the ground?

How long will it take the missile to hit the ground?

Answer 1

h(t) = -16t^2 +400t +100

max h when h'(t) = 0 = -32t+400

t = 400/32 = 25/2 = 12 1/2 seconds when it reaches maximum height and zero velocity

it hits ground at 12 1/2 x 2 = 25 seconds

max h = h(25/2) = -16(25/2)^2 +400(25/2) + 100 = -4(625) +5000 + 100 = 2600 feet

it reaches 2500 feet high twice, once on the way up and 2nd time on the way down

before and after 12 1/2 seconds

h= 2500 = -16t^2 +400t +100

-16t^2 +400t -2400 = 0

t^2 -400t/16 +2400/16 = 0

t^2-25t +100 = 0

t = 25/2 + or - .5sqr(625 -400)

= 12 1/2 + or - 7 1/2

= 5 or 20 seconds

Answer 2

At what time will the missile reach its maximum height? Need to find the x coordinate of the vertex. This is given by the formula:

x=-b/(2a) = -400/[(2)(-16)] = 12.5 sec

What is the maximum height of the missile? Need to find the y coordinate of the vertex. Plug in 12.5 into the equation.

h(t)=-16(12.5)²+400(12.5)+100 = 2600 ft

When will the missile reach 2,500 feet above the ground? Set h(t) equal to 2500 and solve

2500=-16(t)²+400(t)+100 Subtract 2500 from both sides

0=-16(t)²+400(t)-2400

Solving the quadratic equation t=10 seconds and t=15 seconds

How long will it take the missile to hit the ground? Set h(t) equal 0 and solve

0=-16(t)²+400(t)+100

Solving the quadratic equation t=25.2 seconds (the negative solution is not valid)