Time to reach target in exponential model

3. The area, \(A \mathrm {~mm} ^ { 2 }\), of a bacterial culture growing in milk, \(t\) hours after midday, is given by $$A = 20 \mathrm { e } ^ { 1.5 t } , \quad t \geqslant 0$$

1. Write down the area of the culture at midday.
2. Find the time at which the area of the culture is twice its area at midday. Give your answer to the nearest minute.

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 \(t = 55\).

1. A scientist is studying the number of bees and the number of wasps on an island.

The number of bees, measured in thousands, \(N _ { b }\), is modelled by the equation $$N _ { b } = 45 + 220 \mathrm { e } ^ { 0.05 t }$$ where \(t\) is the number of years from the start of the study.
According to the model,

1. find the number of bees at the start of the study,
2. show that, exactly 10 years after the start of the study, the number of bees was increasing at a rate of approximately 18 thousand per year. The number of wasps, measured in thousands, \(N _ { w }\), is modelled by the equation $$N _ { w } = 10 + 800 \mathrm { e } ^ { - 0.05 t }$$ where \(t\) is the number of years from the start of the study.
   When \(t = T\), according to the models, there are an equal number of bees and wasps.
3. Find the value of \(T\) to 2 decimal places.

1. During one day, a biological culure is allowed to grow under controlled conditions.

At 8 a.m. the culture is estimated to contain 20000 bacteria. A model of the growth of the culture assumes that \(t\) hours after 8 a.m., the number of bacteria present, \(N\), is given by $$N = 20000 \times ( 1.06 ) ^ { t } .$$ Using this model,

1. find the number of bacteria present at 11 a.m.,
2. find, to the nearest minute, the time when the initial number of bacteria will have doubled.

6 A number of cases of the general exponential model for the marathon are given in Table 6. One of these is $$R = 115 + ( 175 - 115 ) \mathrm { e } ^ { - 0.0467 t ^ { 0.797 } }$$

1. What is the value of \(t\) for the year 2012?
2. What record time does this model predict for the year 2012?
3. \(\_\_\_\_\)
4. \(\_\_\_\_\)

10 A scientist is researching the effects of caffeine. She models the mass of caffeine in the body using $$m = m _ { 0 } \mathrm { e } ^ { - k t }$$ where \(m _ { 0 }\) 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.
10

1. 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.
   10
2. The scientist wants the mass of caffeine in her body to stay below 480 mg

   10 (b)

   Use the model to find the earliest time

   coffee.

   Give your answer to the nearest minute

   □

A hot drink when first made has a temperature which is \(65°C\) higher than room temperature. The temperature difference, \(d °C\), between the drink and its surroundings decreases by \(1.7\%\) each minute.

1. Show that 3 minutes after the drink is made, \(d = 61.7\) to 3 significant figures. [2]
2. Write down an expression for the value of \(d\) at time \(n\) minutes after the drink is made, where \(n\) is an integer. [1]
3. Show that when \(d < 3\), \(n\) must satisfy the inequality $$n > \frac{\log_{10} 3 - \log_{10} 65}{\log_{10} 0.983}.$$ Hence find the least integer value of \(n\) for which \(d < 3\). [4]
4. The temperature difference at any time \(t\) minutes after the drink is made can also be expressed as \(d = 65 \times 10^{-kt}\), for some constant \(k\). Use the value of \(d\) for 1 minute after the drink is made to calculate the value of \(k\). Hence find the temperature difference 25.3 minutes after the drink is made. [4]

Diane is given an injection that combines two drugs, Antiflu and Coldcure. At time \(t\) hours after the injection, the concentration of Antiflu in Diane's bloodstream is \(3e^{-0.02t}\) units and the concentration of Coldcure is \(5e^{-0.07t}\) units. Each drug becomes ineffective when its concentration falls below 1 unit.

1. Show that Coldcure becomes ineffective before Antiflu. [3]
2. Sketch, on the same diagram, the graphs of concentration against time for each drug. [2]
3. 20 hours after the first injection, Diane is given a second injection. Determine the concentration of Coldcure 10 hours later. [2]