Topic 2.5: Exponential and Sinusoidal Models Questions

The height of the tide h metres at a harbour is modelled by h(t) = 3 sin(πt/6) + 5, where t is hours after midnight. (a) Write down the amplitude of h. (b) Find the period of h. (c) Find the maximum height of the tide.

A taxi charges a fixed fee of $3.50 plus $1.80 per kilometre. (a) Write a linear model for the total cost C ($) in terms of distance d (km). (b) Find the cost of a 12 km trip. (c) Interpret the gradient in context.

A ball is thrown upward. Its height h metres after t seconds is h(t) = −4.9t² + 14.7t + 1. (a) Find the maximum height. (b) Find the time when the ball hits the ground. Give your answer to 3 significant figures.

The value V ($) of a car t years after purchase is modelled by V = 24000 · 0.82ᵗ. (a) Write down the initial value of the car. (b) Find the value after 4 years. (c) Find the percentage decrease in value from year 0 to year 4.

The speed S (m/s) of a river is modelled as S = 1.4d^0.6, where d is the depth (m). (a) Find the speed when d = 5. (b) Find the depth that gives a speed of 3 m/s. Give your answer to 3 s.f.

Water is draining from a tank. The volume V (litres) after t minutes is V = −12t + 240. (a) Find the initial volume. (b) Find the time when the tank is empty. (c) State whether the model is valid for t = 25. Give a reason.

The height (in cm) of a point on a Ferris wheel is h(t) = 10 sin(πt/30) + 12, where t is seconds. (a) Find the height when t = 15. (b) Find the first time when the height is 22 cm.

f(t) = a sin(bt) + d. The graph has a maximum of 9, a minimum of 1, and a period of 8. Find the values of a, b, and d.

The temperature T (°C) in a city varies during the day. The maximum temperature is 28°C and the minimum is 12°C. The maximum occurs at 14:00 and the period is 24 hours. (a) Find the amplitude. (b) Find the midline (vertical shift d). (c) Write a sinusoidal model for T in terms of hours t after midnight.

The depth of water in a harbour follows a sinusoidal pattern. At high tide (t = 2 h after midnight) the depth is 6.5 m. At low tide (t = 8 h) the depth is 1.5 m. (a) Find the amplitude and midline. (b) Find the period. (c) Write a model for the depth D at time t hours after midnight.

At time t = 0, a town has population 12 000. Two years later the population is 13 200. Assuming exponential growth, find the annual growth rate as a percentage.

A bacterial culture starts with 500 bacteria and doubles every 3 hours. (a) Write an exponential model for the number of bacteria N after t hours. (b) Find the number of bacteria after 9 hours. (c) Find the time when the number of bacteria first exceeds 10 000.