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

## 1. | GCSE Higher |

Richard studies fish populations in a lake.

The size of a certain fish population at the start of 2015 was 3700.

The size of this fish population increases exponentially.

Richard assumes that the rate of increase is 15% per year.

(a) Using his assumption, work out the size of this fish population at the start of 2010.

(b) Richard’s assumption is too high. Explain how your answer to part (a) is affected.

## 2. | IB Studies |

While studying a new disease, scientists found that the number of toxic cells in the bloodstream increased over time, according to the model \(D(t)=12×(1.16)^t , t \ge 0\) where \(D\) is the number of the toxic cells in the bloodstream per litre and \(t\) is the time in hours.

(a) Find the number of toxic cells in the bloodstream at \(t=0\).

(b) Calculate the number of toxic cells in the bloodstream after 3 hours.

(c) Determine the time it takes for the number of toxic cells in the bloodstream to first exceed to 200 per litre. Give your answer to the nearest minute.

## 3. | IB Studies |

The diagram show a wheelchair ramp from ground level to a height of 70cm.

(a) Calculate the gradient of the ramp.

Ramp manufacturers recommend a gradient no steeper than 1:12 for safe independent use.

(b) Using your answer to part (a), comment on whether this wheelchair ramp is safe for independent use.

The equation of a different wheelchair ramp is \(2x + 22y -1860 = 0\) where \(y\) represents the height and \(x\) the horizontal distance, both in centimetres.

(c) Find the horizontal distance covered by this wheelchair ramp.

(d) Determine whether this wheelchair ramp is safe for independent use.

## 4. | IB Standard |

Percy Cod and Fran Finklestein are both researchers working at different universities. They are each studying a different colony of bacteria which coincidentally start increasing in size at the same time.

The number of bacteria in Percy's colony, after \(t\) hours, is modelled by the function \(P(t)=8e^{0.3t}\).

(a) Find the initial number of bacteria in Percy's colony.

(b) Find the number of bacteria in Percy's colony after four hours.

(c) How long does it take for the number of bacteria in Percy's colony to reach 350?

The number of bacteria in Fran's colony, after t hours, is modelled by the function \(F(t)=16e^{kt}\).

(d) After four hours, there are 35 bacteria in Fran's colony. Find the value of \(k\).

(e) The number of bacteria in Percy's colony first exceeds the number of bacteria in Fran's colony after \(n\) hours, where \(n\in \mathbf Z\). Find the value of \(n\).

## 5. | IB Studies |

The cross-section of a fish pond is drawn on a set of axes shown below. The edge is modelled by \(y=ax^2+c\) and the cross section is the same for the whole of its length. The curve touches the x-axis at the origin.

Point A has coordinates (-9,5.4) and point B has coordinates (9,5.4).

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

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

(c) Hence write down the equation of the quadratic function which models the edge of the fish pond.

(d) Calculate the value of \(y\) when \(x\)=7.2m.

(e) State what the value of \(x\) and the value of \(y\) represent for this fish pond.

(f) Find the value of \(x\) when the height of water in the pond is 2.7m.

The pond is filled to a maximum depth of 2.7m and the the cross-sectional area of the water is 22.9m^{2}. The pond has a length of 8m.

(g) Calculate the volume of water in the pond.

## 6. | IGCSE Extended |

The diagrams above show a growing fractal of triangles. The sides of the largest equilateral triangle in each diagram are of length 1 metre.

In the second diagram there are four triangles each with sides of length \(\frac{1}{2}\) metre.

In the third diagram there are 16 triangles each with sides of length \(\frac{1}{4}\) metre.

(a) Complete this table for more diagrams.

Diagram 1 | Diagram 2 | Diagram 3 | Diagram 4 | Diagram 5 | Diagram 6 | Diagram \(n\) | ||

Length of Side | 1 | \(\frac{1}{2}\) | \(\frac{1}{4}\) | |||||

Power of 2 | 2^{0} |
2^{-1} |
2^{-2} |

(b) Complete this table for the number of the smallest triangles in diagrams 4, 5 and 6.

Diagram 1 | Diagram 2 | Diagram 3 | Diagram 4 | Diagram 5 | Diagram 6 | Diagram \(n\) | ||

Number of smallest triangles | 1 | 4 | 16 | |||||

Power of 2 | 2^{0} |
2^{2} |
2^{4} |

(c) Calculate the number of the smallest triangles in the diagram where the smallest triangles have sides of length \(\frac{1}{256}\) metre.

## 7. | GCSE Higher |

The quantity of heat required to heat an amount of water is given by the formula:

$$H = atI^2 – b$$Where \(H\) is the number of calories delivered by an electric current of \(I\) amps acting for \(t\) seconds and \(a\) and \(b\) are constants.

(a) Rearrange the formula to make \(I\) the subject.

The graph below gives information about the cooling of a cup of coffee on a cold day. The vertical axes shows the variation in the temperature, \(T\), and the horizontal axis shows the time, \(t\), in seconds.

(b) Work out the average rate of decrease of the temperature of the coffee between \(t = 0\) and \(t = 700\).

The instantaneous rate of decrease of the temperature of the water at time \(A\) seconds is equal to the average rate of decrease of the temperature of the water between \(t = 0\) and \(t = 700\).

(c) Find an estimate for the value of \(A\) showing how you got your answer.

## 8. | IB Studies |

A cup of hot tea is placed on a table to cool. The temperature of the room is 24^{o}C.

This situation can be modelled by the exponential function:

$$T = a + b (k^{-m})$$where \(T\) is the temperature of the water, in ^{o}C, and \(m\) is the number of minutes for which the cup has been placed on the table.

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

Initially the temperature of the tea is 92^{o}C.

(b) Find the value of b.

After one minute, the temperature of the tea is 74^{o}C.

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

(d) Find the temperature of the tea three minutes after it has been placed on the table.

(e) Find the total time needed for the tea to reach a temperature ofof 30^{o}C. Give your answer in minutes and seconds, correct to the nearest second.

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