smc-973-40-Applied Context

The diagram illustrates the design for part of a roller-coaster track. The section `RO` is a straight line with slope 1.2, and the section `PQ` is a straight line with slope – 1.8. The section `OP` is a parabola `y = ax^2 + bx`. The horizontal distance from the `y`-axis to `P` is 30 m.

In order that the ride is smooth, the straight line sections must be tangent to the parabola at `O` and at `P`.

Find the values of `a` and `b` so that the ride is smooth. (3 marks)

--- 6 WORK AREA LINES (style=lined) ---
Find the distance `d`, from the vertex of the parabola to the horizontal line through `P`, as shown on the diagram. (2 marks)

--- 5 WORK AREA LINES (style=lined) ---

Show Answers Only

`text(For a smooth ride,)\ a = -0.05\ text(and)\ b = 1.2`
`16.2\ text(m)`

Show Worked Solution

i.	`y`	`= ax^2 + bx`
	`dy/dx`	`= 2ax + b`
	`text(At)\ \ x = 0,`
	`dy/dx`	`= b`

♦♦♦ Exact data unavailable.
MARKER’S COMMENT: Successful students equated the curve gradient to the straight section as a requirement for a smooth ride.

`text(We need)\ m\ text(at)\ O = 1.2`

`:.\ b = 1.2`

`text(At)\ P,\ x = 30`

`dy/dx`	`= 2 xx a xx 30 + 1.2`
	`= 60a + 1.2`

`text(We need)\ m\ text(at)\ P = -1.8`

`60a + 1.2`	`= -1.8`
`60a`	`= -3`
`a`	`= -3/60`
	`= -0.05`

`:.\ text(For a smooth ride,)\ a = –0.05\ \ text(and)\ \ b = 1.2`

ii.	`y`	`= -0.05x^2 + 1.2x`
	`dy/dx`	`= -0.1x + 1.2`

`text(Find)\ x\ text(when)\ dy/dx = 0`

`-0.1x + 1.2`	`= 0`
`x`	`= 1.2/0.1`
	`= 12`

`:.\ text(MAX when)\ x = 12`

`text(When)\ x = 12`

`y`	`= -0.05 xx 12^2 + 1.2 xx 12`
	`= -7.2 + 14.4`
	`= 7.2`

`text(When)\ x = 30`

`y`	`= -0.05 xx 30^2 + 1.2 xx 30`
	`= -45 + 36`
	`= -9`

`:.\ d`	`= 7.2 + \|–9\|`
	`= 16.2\ text(m)`