Quadratics

Question

The path of a soccer ball can be defined by the relation h= -0.5d^2 + d, where h represents the height in metres and d represents the horizontal distance in meters the ball travels before it hits the ground 

 

a) what is the maximum height of the soccer ball  ? 

b) how far will the ball travel horizontally when the ball hits the ground 

c) What is the height of the soccer ball when it is 7m away from the player the kicked it ? 

 

 

Answer 1

a)

 

To find the maximum height, we need to find the vertex of function h.  The vertex is (L, k).  We can find L using this formula

 

L = -b / 2a

k = h(L)

 

where:

a = -0.5

b = 1

c = 0

 

L = -1 / (2*-0.5)

L = -1 / -1

L = 1

 

Now we evaluate h when d=1.

 

h = (-0.5)(1)² + 1

h = -0.5 + 1

h = 0.5

 

 

The maximum height is 0.5 meters.

 

 

b)

 

We set h=0 and solve for d.

 

0 = -0.5d² + d

 

0 = d(-0.5d + 1)

 

 

Set the factors equal to 0.

 

d = 0             and           -(1/2)d + 1 = 0

 

                                              d = 2

 

 

Reject   d=0 , since this is your starting distance.

 

The ball will reach a horizontal distance of 2 meters when the ball hits the ground.

 

 

c)

 

Set d=7 and evaluate h.

 

h = -0.5(7)² + 7

 

h = -0.5(49) + 7

 

h = -24.5 + 7

 

h = -17.5

 

 

We cannot have a negative height.  Either the value of d must be smaller, or your function of h must be modified.