8) a.

TASK

An Intense Source of Light

Photographers often use multiple light sources to control shadows or to

illuminate their subject in an artistic manner. The illumination from a light

source is inversely proportional to the square of the intensity of the source.

Two light sources, L₁ and La, are 10 m apart, and have intensity of 4 units and

1 unit, respectively. The illumination at any point P is the sum of the

illuminations from the two sources.

Hint: When a quantity y is inversely proportional to the square of another

quantity x, then y, where k is a constant.

10 m

a) Determine a function that represents the illumination at the point P relative

to the distance from L₁.

b) At what distance from L, is the illumination on P the greatest?

At what distance from L, is the illumination on P the least?

Investigate whether it would be more effective to increase the intensity of

L₂ or to move L₂ closer to L, in order to increase the illumination at the

point found in part b). Support your findings with mathematical evidence.

Consider both the illumination and the rate of change of the illumination

as L₂ changes intensity or position.

b.

TASK

Double Ferris Wheel

Some amusement parks have a double Ferris wheel, which consists of two

vertically rotating wheels that are attached to each other by a bar that also

rotates. There are eight gondolas equally spaced on each wheel. Riders

experience a combination of two circular motions that provide a sensation

more thrilling than the classic single Ferris wheel. In particular, riders

experience the greatest sensation when their rate of change in height is the

greatest.

. Each of the two wheels is 6 m in

diameter and revolves every 12 s

. The rotating bar is 9 m long. The

ends of the bar are attached to the

centres of the wheels.

• The height from the ground to

the centre of the bar is 8 m. The

bar makes a complete revolution

every 20 s.

A rider starts seated at the

lowest position and moves

counterclockwise.

. The bar starts in the vertical

position.

Consider the height of a rider who begins the ride in the lowest car.

a) Write a function f(t) that expresses the height of the rider relative to the

centre of the wheel at time t seconds after the ride starts. Write a second

function g (f) that expresses the position of the end of the bar (the centre of

the rider's wheel) relative to the ground at time t seconds.

b) Explain how the sum of these two functions gives the rider's height above

the ground after t seconds.

Use technology to graph the two functions and their sum for a 2-min ride.

What is the maximum height reached by the rider? When does this occur?

) What is the maximum vertical speed of the rider? When does this occur?

f) Design your own double Ferris wheel. Determine the position function for

a rider on your wheel. What is the maximum speed experienced by your

riders? Is there a simple relationship between the dimensions of the Ferris

wheel and the maximum heights or speeds experienced?