quadratic equation word problem

Question

A diver jumps off a diving board that is 10 ft above the water at a velocity of 12 ft/sec. His height, s , in feet, above the water can be modeled by s(t)=-16t^2+12t+10.
(a) How long is the diver in the air before he hits the water?
Enter the exact answer.
The diver is in the air for ____ seconds.
(b) What is the maximum height achieved and when does it occur?
Enter the exact answers.
The maximum height is ____ feet, and it occurs after ____seconds.

Answer 1

(a) How long is the diver in the air before he hits the water? You need to find t when s(t) = 0.  

You can use the graphing calculator or solve algebraically by setting the equation to zero.

 

-16t² + 12t + 10 = 0 

 

Factor and solve for t

 

-16t² + 20t - 8t + 10 = 0

 

- 4t (4t - 5t) - 2 (4t - 5) = 0 

 

(4t - 5) (-4t - 2) = 0 

 

(4t - 5) = 0 or  (-4t - 2) = 0

 

4t = 5         or        -4t = 2

 

t = 5/4        or         t = -2/4 = -1/2

 

t = 5/4 = 1.25.  

 

The diver is in the air for 1.25 seconds before he hits the water.

 

 

(b) What is the maximum height achieved and when does it occur?

 

You need to find the vertex.  

 

First find the line of symmetry using the formula t = -b/2a

 

Then, substitute that t value in the equation to find the maximum height.

 

t = -b/2a

 

t = -12/2(-16)

 

t = -12/-32

 

t = 3/8 = 0.375 ------------ line of symmetry

 

substitute  t = 3/8 in the equation to find the maximum height.

 

s (3/8) = -16t² + 12t + 10

 

s (3/8) = -16 (3/8)² + 12 (3/8) + 10

 

s (3/8) = -16 (9/64) + 12 (3/8) + 10

 

s (3/8) = - (9/4) +  (9/2) + 10

 

s (3/8) = (- 9 + 18 + 40) / 4

 

s (3/8) = 49/4 

 

s (3/8) = 12.2 ft

 

Vertex : (0.375, 12.2)

 

The maximum height is 12.2 feet, and it occurs after 0.375 seconds.