Exponential growth and decay

A gardener has a greenhouse containing 900 tomato plants.

The gardener notices that some of the tomato plants are damaged by insects.

Initially there are 25 damaged tomato plants.

The number of tomato plants damaged by insects is increasing by 32% each day.

The total number of plants damaged by insects, x, is modelled by

where A and B are constants and t is the number of days after the gardener first noticed the damaged plants.

Use this model to find the total number of plants damaged by insects 5 days after the gardener noticed the damaged plants.

Explain why this model is not realistic in the long term.

A refined model assumes the rate of increase of the number of plants damaged by insects is given by

Show that

where A and B are positive integers to be found.

Hence, find t in terms of x.

Hence, find the number of days it takes from when the damage is first noticed until half of the plants are damaged by the insects.

Section B

all questions in the spaces provided.

The number of radioactive atoms, N, in a sample of a sodium isotope after time t hours can be modelled by

where N₀ is the initial number of radioactive atoms in the sample and k is a positive constant.

The model remains valid for large numbers of atoms.

It takes 15.9 hours for half of the sodium atoms to decay.

Determine the number of days required for at least 90% of the number of atoms in the original sample to decay.

Find the percentage of the atoms remaining after the first week.

Give your answer to two significant figures.

Explain why the model can only provide an estimate for the number of remaining atoms.

Explain why the model is invalid in the long run.

Theresa bought a house on 2 January 1970 for £8000.

The house was valued by a local estate agent on the same date every 10 years up to 2010.

The valuations are shown in the following table.

The valuation price of the house can be modelled by the equation

where V pounds is the valuation price t years after 2 January 1970 and p and q are constants.

Show that V pq^t can be written as log₁₀ V log₁₀ p t log₁₀ q

The values in the table of log 10 V against t have been plotted and a line of best fit has been drawn on the graph below.

Using the given line of best fit, find estimates for the values of p and q.

Give your answers correct to three significant figures.

Determine the year in which Theresa’s house will first be worth half a million pounds.

Explain whether your answer to part (c) is likely to be reliable.

An elite athlete runs in a straight line to complete a 100-metre race.

where t is the time, in seconds, after the starting pistol is fired.

Find the maximum value of v, giving your answer to one decimal place.

Fully justify your answer.

Find an expression for the distance run in terms of t.

The athlete’s actual time for this race is 9.8 seconds.

Comment on the accuracy of the model.

A student is conducting an experiment in a laboratory to investigate how quickly liquids cool to room temperature.

A beaker containing a hot liquid at an initial temperature of 75 C cools so that the temperature, C, of the liquid at time t minutes can be modelled by the equation

After 2 minutes the temperature falls to 68 C.

Find the temperature of the liquid after 15 minutes.

Give your answer to three significant figures.

Find the room temperature of the laboratory, giving a reason for your answer.

Find the time taken in minutes for the liquid to cool to 1 C above the room temperature of the laboratory.

Explain why the model might need to be changed if the experiment was conducted in a different place.

A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using

where m₀ milligrams is the initial mass of caffeine in the body and m milligrams is the mass of caffeine in the body after t hours.

On average, it takes 5.7 hours for the mass of caffeine in the body to halve.

One cup of strong coffee contains 200 mg of caffeine.

The scientist drinks two strong cups of coffee at 8 am. Use the model to estimate the mass of caffeine in the scientist’s body at midday.

The scientist wants the mass of caffeine in her body to stay below 480 mg

Use the model to find the earliest time that she could drink another cup of strong coffee.

Give your answer to the nearest minute.

State a reason why the mass of caffeine remaining in the scientist’s body predicted by the model may not be accurate.