The path that a diver follows is given by y = -0.4(x-4)^2 + 14 where x is the horizontal distance (in feet) from the edge of the diving board and y is the height above the water (in feet).

Question

a.) What is the maximum height of the diver?

b.) At what horizontal distance do they reach the max height?

c.) At what horizontal distance do they enter the water?

d.) How high is the diving board?

Answer 1

y = -0.4(x - 4)² + 14

Equation is vertex form for a parabola. y = a(x - h)² + k, where a is constant that affects the shape and

direction of the curve, h and k are the ordered (x, y) pair for the vertex respectively.

So the vertex will be at (4, 14).

a) Maximum height of diver is 14 ft.

b) Horizontal. distance from board at max. ht. is 4 ft.

c) Horizontal distance when enter water is where y = 0.

0 = -0.4(x - 4)² + 14

-14/-0.4 = (x - 4)² = 35

x - 4 = ±√35

= ±5.92 + 4 = 9.92 ft (only the positive value makes sense).

d) Height of board is where x = 0.

y = -0.4(0 - 4)² + 14 → -0.4(16) + 14 → -6.4 + 14 = 7.6 ft.

Suggest you use Desmos Graphing Calculator to verify these answers.

Answer 2

y=-0.4x^2+3.2x+7.6

the vertex is (m,n) m=-b/2a=-3.2/2(-0.4)=4

1) The maximum height is : f(4)=-0.4x16+3.2x4+7.6=-6.4+12.8+7.6=14m

2)At what horizontal distance do they reach the max height? at the distance 4 m they reach the maximum height

f(4)=14 m .

3)At what horizontal distance do they enter the water? We have to solve the equation -0.4x^2+3.2x+7.6=0

a=-0.4,b=3.2 ,c=7.6 D=b^2-4ac=(3.2)^2-4x(-0.4)x(7.6)=22.4 √D=4.7 x=(-b±√D)/2a=(-3.2 ± 4.7)/-0.8

x=-1.9m (we cannot accept negative length) so at the distance x=9.88m they entered .

4)How high is the diving board? This is the starting point y=7.6 m

Answer 3

a) y – 14 = –2(x – 4)²/5 ≤ 0 so y ≤ 14.

b) y(4) = 14.

c) y → 0 at the positive root of 0 = 14 – 2(x – 4)²/5, so as x → 4 + √35, between 9 and 10 ft.

d) y(0) = 14 – 2×4²/5 = 38/5 = 7.6 ft.