Rate of change in exponential model

2. The current, \(I\) amps, in an electric circuit at time \(t\) seconds is given by $$I = 16 - 16 ( 0.5 ) ^ { t } , \quad t \geqslant 0$$ Use differentiation to find the value of \(\frac { \mathrm { d } I } { \mathrm {~d} t }\) when \(t = 3\).
Give your answer in the form \(\ln a\), where \(a\) is a constant.

1. The number of bacteria, \(N\), present in a liquid culture at time \(t\) hours after the start of a scientific study is modelled by the equation

$$N = 5000 ( 1.04 ) ^ { t } , \quad t \geqslant 0$$ where \(N\) is a continuous function of \(t\).

1. Find the number of bacteria present at the start of the scientific study.
2. Find the percentage increase in the number of bacteria present from \(t = 0\) to \(t = 2\) Given that \(N = 15000\) when \(t = T\),
3. find the value of \(\frac { \mathrm { d } N } { \mathrm {~d} t }\) when \(t = T\), giving your answer to 3 significant figures.

1. The prices of two precious metals are being monitored.

The price per gram of metal \(A , \pounds V _ { A }\), is modelled by the equation $$V _ { A } = 100 + 20 \mathrm { e } ^ { 0.04 t }$$ where \(t\) is the number of months after monitoring began.
The price per gram of metal \(B , \pounds V _ { B }\), is modelled by the equation $$V _ { B } = p \mathrm { e } ^ { - 0.02 t }$$ where \(p\) is a positive constant and \(t\) is the number of months after monitoring began.
Given that \(V _ { B } = 2 V _ { A }\) when \(t = 0\)

1. find the value of \(p\) When \(t = T\), the rate of increase in the price per gram of metal \(A\) was equal to the rate of decrease in the price per gram of metal \(B\)
2. Find the value of \(T\), giving your answer to one decimal place.
   (Solutions based entirely on calculator technology are not acceptable.)

1. The growth of pond weed on the surface of a pond is being investigated.

The surface area of the pond covered by the weed, \(A \mathrm {~m} ^ { 2 }\), can be modelled by the equation $$A = 0.2 \mathrm { e } ^ { 0.3 t }$$ where \(t\) is the number of days after the start of the investigation.

1. State the surface area of the pond covered by the weed at the start of the investigation.
2. Find the rate of increase of the surface area of the pond covered by the weed, in \(\mathrm { m } ^ { 2 } /\) day, exactly 5 days after the start of the investigation. Given that the pond has a surface area of \(100 \mathrm {~m} ^ { 2 }\),
3. find, to the nearest hour, the time taken, according to the model, for the surface of the pond to be fully covered by the weed. The pond is observed for one month and by the end of the month \(90 \%\) of the surface area of the pond was covered by the weed.
4. Evaluate the model in light of this information, giving a reason for your answer.

5. The mass, \(m\) grams, of a radioactive substance, \(t\) years after first being observed, is modelled by the equation $$m = 25 \mathrm { e } ^ { - 0.05 t }$$ According to the model,

1. find the mass of the radioactive substance six months after it was first observed,
2. show that \(\frac { \mathrm { d } m } { \mathrm {~d} t } = k m\), where \(k\) is a constant to be found.

10 In a certain region, the populations of grey squirrels, \(P _ { \mathrm { G } }\) and red squirrels \(P _ { \mathrm { R } }\), at time \(t\) years are modelled by the equations: \(P _ { \mathrm { G } } = 10000 \left( 1 - \mathrm { e } ^ { - k t } \right)\) \(P _ { \mathrm { R } } = 20000 \mathrm { e } ^ { - k t }\) where \(t \geq 0\) and \(k\) is a positive constant.

1. 1. On the axes in your Printed Answer Book, sketch the graphs of \(P _ { \mathrm { G } }\) and \(P _ { \mathrm { R } }\) on the same axes.
   2. Give the equations of any asymptotes.
2. What does the model predict about the long term population of
   • grey squirrels
   • red squirrels?
   Grey squirrels and red squirrels compete for food and space. Grey squirrels are larger and more successful than red squirrels.
3. Comment on the validity of the model given by the equations, giving a reason for your answer.
4. Show that, according to the model, the rate of decrease of the population of red squirrels is always double the rate of increase of the population of grey squirrels.
5. When \(t = 3\), the numbers of grey and red squirrels are equal. Find the value of \(k\).

3 The mass, \(m\) grams, of a substance at time \(t\) years is given by the formula $$m = 180 \mathrm { e } ^ { - 0.017 t }$$

1. Find the value of \(t\) for which the mass is 25 grams.
2. Find the rate at which the mass is decreasing when \(t = 55\).