## Exam-Style Questions on Circular functions## Problems on Circular functions adapted from questions set in previous Mathematics exams. |

## 1. | IB Standard |

Consider the graph of \(f(x)=a\sin(b(x+c))+12\), for \(0\le x\le 24\).

The graph has a maximum at (8, 22) and a minimum at (18, 2).

(a) Find the value of \(a\).

(b) Find the value of \(b\).

(c) Find the value of \(c\).

(d) Solve \(f(x)=5\).

## 2. | IB Standard |

The diagram shows part of the graph of \(y=a\sin{(bx)}+c\) with a minimum at \((-2.5,-2)\) and a maximum at \((2.5,4)\).

(a) Find \(a\).

(b) Find \(b\).

(c) Find \(c\).

## 3. | IB Standard |

The Big Wheel at Fantasy Fun Fayre rotates clockwise at a constant speed completing 15 rotations every hour. The wheel has a diameter of 90 metres and the bottom of the wheel is 6 metres above the ground.

A cabin starts at the bottom of the wheel with the top of the cabin 6m above the ground.

(a) Find the greatest height of the top of the cabin reaches as the wheel rotates.

After \(t\) minutes, the height \(h(t)\) metres above the ground of the top of a cabin is given by the function \(h(t)=51-a\cos bt\).

(b) Find the period of \(h(t)\)

(c) Find the value of \(b\).

(d) Find the value of \(a\).

(e) Sketch the graph of \(h(t)\) , for \(0\le t\le 5\).

(f) In one rotation of the wheel, find the probability that a randomly selected seat is at least 70 metres above the ground. Give your answer to two decimal places.

## 4. | IB Standard |

The population of sheep on a ranch is modelled by the function \(P(t)= 65 \sin(0.4t-32)+450\), where t is in months, and \(t=1\) corresponds to 1st January 2015.

(a) Find the number of sheep on the ranch on 1st July 2015.

(b) Find the rate of change of the sheep population on 1st July 2015.

(c) Explain your answer to part (b) with reference to the sheep population size on 1st July2015.

## 5. | IB Analysis and Approaches |

The Fun Wheel at the Meller Theme Park rotates at a constant speed.

- The lowest point on the wheel is 7 metres above ground.
- The diameter of the wheel is 50 metres.
- The wheel takes 4 minutes to make one revolution.

The height, \(h\) metres, of the point initially at the top of the circumference of the wheel after \(t\) minutes is given by:

$$ h(t) = a \cos(bt) + c $$(a) Find the values of \(a, b \text{ and } c \text{ where } a,b,c \in \mathbb{R} \).

(b) Draw a sketch of the function \(h(t)\) for \(0 < t < 4 \).

## 6. | IB Standard |

A Big Wheel at an amusement park has a diameter of length 70 metres which rotates at a constant speed. The bottom of the wheel is h metres above the ground. A seat starts at the bottom of the wheel.The wheel completes one revolution in 4 minutes. [The diagram is not to scale]

(a) After 2 minutes, the seat is 76m above the ground. Find h.

(b) After t minutes, the height of the seat above ground is given by \(f(t) =41+ k \cos{ \frac{\pi t}{2}} \) for \(0 \le t \le 16 \).

Find the value of k.

(c) Find when the seat is 40 m above the ground for the third time.

## 7. | IB Standard |

Let \(f(x)=\sin ( \frac {\pi}{4}x) + \cos ( \frac {\pi}{4}x) \), for \(-4\le x \le 4\)

(a) Sketch the graph of \(f\).

(b) Find the values of \(x\) where the function is decreasing.

(c) The function \(f\) can also be written in the form \(f(x)=a\sin ( \frac {\pi}{4}(x+c))\) where \(a\in \mathbb R\) and \(0 \le c \le 2\). Find the value of \(a\) and \(c\).

## 8. | A-Level |

The height above the ground, H metres, of a passenger on a Ferris wheel t minutes after the wheel starts turning, is modelled by the following equation:

$$H = k – 8\cos (60t)° + 5\sin (60t)°$$where k is a constant.

(a) Express \(H\) in the form \(H = k - R \cos(60t + a)° \) where \(R\) and \(a\) are constants to be found (\( 0° \lt a \lt 90° \)).

(b) Given that the initial height of the passenger above the ground is 2 metres, find a complete equation for the model.

(c) Hence find the maximum height of the passenger above the ground.

(d) Find the time taken for the passenger to reach the maximum height on the fifth cycle. (Solutions based entirely on graphical or numerical methods are not acceptable.)

(e) It is decided that, to increase profits, the speed of the wheel is to be increased. How would you adapt the equation of the model to reflect this increase in speed?

## 9. | IB Analysis and Approaches |

The height of water in metres above Chart Datum* in the Port of Bristol on a day in January is modelled by the function:

$$H(x) = a \sin(b(x-c)) +d$$where \(x\) is the number of hours after midnight GMT, and \(a,b,c, \text{ and } d \) are constants greater than zero.

The following graph shows the height of the water for 19 hours starting at midnight.

The first low tide occurs at 02:23 and next low tide occurs at 14:28.

Throughout the day the height of the water fluctuates between 1m and 13.8m.

(a) Show that \(a = 6.4\).

(b) Find the value of \(b\) to two decimal places.

(c) Find the value of \(d\).

(d) Find the smallest possible value of \(c\) to one decimal place.

(e) Using the values of \(a,b,c, \text{ and } d \) found above, find the height of the water at 10:00.

(f) Determine the number of hours, during this January day, for which the tide is higher than 10 metres above Chart Datum.

* Chart Datum is a reference datum or level from which heights and depths on a nautical chart are measured. Chart Datum is determined by the lowest astronomical tide, which is the level of the sea at its lowest possible level. It is also the reference level used to measure the depth of water in a harbour or other body of water.

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