Hello, shenton!

The path of a thrown ball can be represented by the relation:

. . h(d) .= .-(1/150)d² + (2/5)d + (3/2)

where d metres is the horizontal distance travelled and h(d) metres is the height.

(a) What is the maximum height reached by the ball?

(b) What horizontal distance has the ball travelled when it reaches its maximum height?

Answer key: (a) 7.5m (b) 30m

The height function is a quadratic; its graph is a down-opening parabola.

. . It reaches its maximum at its vertex.

The vertex of the parabola: y .= .ax² + bx + c

. . is at: .x .= .-b/2a

Our parabola has: a = (-1/150), b = (2/5)

. . Hence: .d .= .-(2/5)/2(-1/150) .= .30

(b) At maximum height, the ball has moved 30 m horizontally.

(a) The maximum height is:

. . h(30) .= .-(1/150)(30²) + (2/5)(30) + 3/2 .= .15/2 .= .7.5 m