Questions

1. Question 1

   Find the maximum and minimum values of the following functions:

   (i)

   $3 - \sin 3x$

   (ii)

   $3 + \sin 2x$

   (iii)

   $\dfrac{1}{2} + \sin(5x + \pi)$

   (iv)

   $\dfrac{3}{2} + \cos\left(x - \dfrac{\pi}{4}\right)$

   (v)

   $1 - 3\cos 2x$

   (vi)

   $1 + 2\sin\left(x + \dfrac{\pi}{6}\right)$

   (vii)

   $\dfrac{1}{10 - 2\sin 3x}$

   (viii)

   $\dfrac{1}{7 + 3\cos(-2x)}$

   (ix)

   $\dfrac{1}{5 - 3\cos(3x - 1)}$

   SolutionTheory
2. Question 2

   The temperature $T$ in degrees Celsius of a certain city varies throughout the day according to the equation $T(t) = \dfrac{13}{2}\sin\left(\dfrac{\pi}{6}t - \dfrac{\pi}{9}\right) + 15$, where $t$ is the time in hours, with $t = 0$ corresponding to midnight.

   (a)

   Find the maximum and minimum temperature during the day.

   (b)

   Find the temperature at $t = 9$ hours (9:00 a.m.).

   SolutionTheory
3. Question 3

   A man on the top of a 100 m high light-house is in line with two ships on the same side of it, whose angles of depression from the man are $17^\circ$ and $19^\circ$ respectively. Find the distance between the ships.

   SolutionTheory
4. Question 4

   $P$ and $Q$ are two points in line with a tree. If the distance between $P$ and $Q$ be 30 m and the angles of elevation of the top of the tree at $P$ and $Q$ are $12^\circ$ and $15^\circ$ respectively, find the height of the tree.

   SolutionTheory
5. Question 5

   A giant Ferris wheel has a diameter of 60 feet. The lowest point of the wheel is located 6 feet above the ground. The wheel completes one full revolution every 80 seconds.

   (a)

   Model an equation that represents the height $h(t)$ of a rider on the Ferris wheel at any given time $t$.

   (b)

   Find the maximum height of the rider.

   (c)

   Find the height of the rider from the ground after 35 seconds.

   SolutionTheory
6. Question 6

   A child is playing on a swing in a playground. The height $h(t)$ of the swing seat above the ground (in metres) at time $t$ (in seconds) is modeled by the function: $h(t) = 1.5 + 1.2\sin(3\pi t)$.

   (a)

   What is the maximum height reached by the swing seat?

   (b)

   What is the minimum height reached by the swing seat?

   (c)

   How long does it take for the swing to complete one full back-and-forth motion (period)?

   (d)

   At what time(s) does the swing seat first reach a height of 2.12 metres?

   SolutionTheory
7. Question 7

   A carnival ride consists of a vertical wheel with a diameter of 40 feet. The centre of the wheel is 28 feet above the ground. The wheel rotates at a constant speed and takes 120 seconds to make one complete revolution. Model an equation that describes the height $h(t)$ of a rider on the wheel as a function of time $t$. How high is the rider from the ground after 90 seconds? At what times will the rider be 36 feet above the ground?

   SolutionTheory
8. Question 8

   Suppose the temperature $T$ in degrees Fahrenheit of Lahore city in a month of December throughout the day can be modeled by the equation: $T = 64 + 8\sin\left(\dfrac{\pi}{12}(t - 8)\right)$, where $t$ is the time in hours. The temperature oscillates 8 degrees above and below an average temperature of 64 degrees.

   (a)

   Find the temperature at $t = 9$ hours.

   (b)

   At what time the temperature will be maximum?

   (c)

   Calculate the maximum temperature.

   SolutionTheory
9. Question 9

   Suppose the population of a coastal city follows a sinusoidal pattern due to seasonal migration. The population of the city over the course of a year can be modeled by the equation: $P(t) = 70000 + 10000\cos\left(\dfrac{\pi}{6}t - \dfrac{\pi}{2}\right)$, $P(t)$ is the population at time $t$ ($t$ is the time in months, with $t = 0$ corresponding to January 1st).

   (a)

   Find the population of the city at $t = 7$ months.

   (b)

   Find the maximum population.

   SolutionTheory