5.5 Data Collecting and Modelling
Chapter
Chapter 5
Section
5.5
Solutions 11 Videos

A motion sensor is used to gather data on the motion of a pendulum. The table of values is exported to a computer, and graphing software is used to draw the graph shown. Time, in seconds, is on the horizontal axis. Distance, in metres, is on the vertical axis.

a) Use the graph to estimate the maximum and minimum values. Then, use these values to find the approximate amplitude, a.

b) Graph and sketch a horizontal reference line. Estimate the vertical shift.

c) Use the horizontal reference line to estimate the phase shift, d.

d) Use the horizontal reference line to estimate the period. Use the period to find the value of k.

e) Construct a model for the motion by writing an equation using a sinusoidal function.

Q2

The height, h, in metres, of the tide in a given location on a given day at t hours after midnight can be modelled using the sinusoidal function h(t) = 5 \sin [30(t - 5)] + 7.

a) Find the maximum and minimum values for the depth, h, of the water.

b) What time is high tide? What time is low tide?

c) What is the depth of the water at 9:00 a.m.?

d) Find all the times during a 24-h period when the depth of the water is 3 In.

Q3

The population, P, of a lakeside town with a large number of seasonal residents can be modelled using the function P(t) = 5000\sin[30(t- 7)] + 8000, where t is the number of months after New Year's day.

(a) Find the maximum and minimum values for the population over a whole year.

(b) When is the population a maximum?

(c) What is the population on September 30?

(d) When is the population about 10 000?

2.43mins
Q4

(a) A sine function has half the period of y = sin x. All other parameters of the two functions are the same.

a) Predict the number of points of intersection if the two functions are graphed from x = 0^o to x = 360^o. Justify your prediction.

b) Determine the number of points of intersection if the two functions are graphed from x = 0^o to x = 360^o. Was your prediction correct? If not, explain why.

c) Determine the coordinates of the first point to the right of the origin where the graphs of the two functions intersect.

3.07mins
Q5

A Ferris wheel has a diameter of 20 m and is 4 m above ground level at its lowest point. Assume that a rider enters a car from a platform that is located 30^{\circ} around the rim before the car reaches lowest point.

• Model the rider's height above the ground vs angle using a transformed sine function.
2.58mins
Q6a

A Ferris wheel has a diameter of 20 m and is 4 m above ground level at its lowest point. Assume that a rider enters a car from a platform that is located 30^{\circ} around the rim before the car reaches lowest point.

• Model the rider's height above the ground versus angle using a transformed cosine function.
0.30mins
Q6b

A Ferris wheel has a diameter of 20 m and is 4 m above ground level at its lowest point. Assume that a rider enters a car from a platform that is located 30^{\circ} around the rim before the car reaches lowest point.

• Suppose that the platform is moved to 60^{\circ} around the rim from the lowest position of the car.
• i) Model the rider's height above the ground vs angle using a transformed sine function.
• ii) Model the rider's height above the ground versus angle using a transformed cosine function.
0.52mins
Q6c

The movement of a piston in an automobile engine can be modelled by the function y = 50\sin 10800t + 20, where y is the distance, in millimetres, from the crankshaft and t is the time, in seconds.

(a) What is the period of the motion?

(b) Determine the maximum, minimum, and amplitude.

(c) When do the maximum and minimum values occur?

(d) What is the vertical position of the piston at t = \frac{1}{120}s?

1.42mins
Q8

Smog is generic term used to describe pollutants in the air. A smog alert is usually issued when the air quality index is greater than 50. Air quality can vary throughout the day, increasing when more cars are on the road. Consider a model of the form I = 30\sin[15(t - 4)] + 25, where I is the value of the air quality index and t measures the time after midnight, in hours.

(a) What is the period of the modelled function? Explain why this makes sense.

(b) Determine the maximum, minimum, and amplitude.

(c) When do the maximum and minimum occur?

(d) During what time interval would a smog alert be issued, according to this model?

3.10mins
Q12abc

Consider a 10 m tall tree. On a certain day, the sun rises at 6:00am, is directly overhead at noon, and set at 6:00pm. From 6:00am until noon, the length, s, in metres, of the trees shadow can be modelled by the relation s = 10 \cot 15 t, where t is the time, in hours, past 6:00am.

(a) Modify the model so that it is valid from noon until 6:00 pm.

(b) Graph the model and the modified model in part (a) on the same se of axes.