Question

Jean threw a disc in the air. The height of the disc can be modelled by the function

h = -5t^2 + 31.5t + 2, where h is the height in metres after t seconds.
Patrick fired a paintball at the disc. The path of the paintball is modelled by the function h = 30t + 1, with the same units. How long will it take the paint ball to hit the disc?

Answer 1

It will take 0.62 seconds for the paint ball to hit the disc.

Explanation:

Height of the disk:

`H_d = -5t^2 + 31.5t + 2`

Height of the paintball:

`H_p = 30t + 1`

When the paintball will hit the disk?

When they are at the same height, so:

`H_d = H_p`

`-5t^2 + 31.5t + 2 = 30t + 1`

`5t^2 - 1.5t - 1 = 0`

Solving a quadratic equation:

Given a second order polynomial expressed by the following equation:

`ax^(2) + bx + c, a\\eq0`.

This polynomial has roots
`x_(1), x_(2)` such that
`ax^(2) + bx + c = a(x - x_(1))*(x - x_(2))`, given by the following formulas:

`x_(1) = (-b + √(\Delta))/(2*a)`

`x_(2) = (-b - √(\Delta))/(2*a)`

`\Delta = b^(2) - 4ac`

In this question:

Quadratic equation with
`a = 5, b = -1.5, c = -1`

So

`\Delta = (-1.5)^2 - 4(5)(-1) = 22.25`

`t_(1) = (-(-1.5) + √(22.25))/(2(5)) = 0.62`

`t_(2) = (-(-1.5) - √(22.25))/(2(5)) = -0.32`

Time is a positive measure, so 0.62.

It will take 0.62 seconds for the paint ball to hit the disc.