Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Differential Equations, Exercise 3: EXERCISE 10B

The half-life of a radioactive isotope is the amount of time it takes for half of the isotope in a sample to decay to its stable form. Carbon$-14$ is a radioactive isotope that has a half-life of $5700$ years. It is given that the rate of decrease of the mass, $m$, of the carbon$-14$ in a sample is proportional to its mass. A sample of carbon$-14$ has initial mass $m_0$. What fraction of the original amount of carbon$-14$ would be present in this sample after $2500$ years?

A bottle of water is taken out of a refrigerator. The temperature of the water in the bottle is $4°\mathrm{C}.$ The bottle of water is taken outside to drink. The air temperature outside is constant at $24°\mathrm{C}.$ It is given that the rate at which the water in the bottle warms up is proportional to the difference in the air temperature outside and the temperature of the water in the bottle. After $2$ minutes the temperature of the water in the bottle is $11°\mathrm{C}.$ How long does it take for the water to warm $20°\mathrm{C}$?

A bottle of water is taken out of a refrigerator. The temperature of the water in the bottle is $4°\mathrm{C}.$ The bottle of water is taken outside to drink. The air temperature outside is constant at $24°\mathrm{C}.$ It is given that the rate at which the water in the bottle warms up is proportional to the difference in the air temperature outside and the temperature of the water in the bottle. After $2$ minutes the temperature of the water in the bottle is $11°\mathrm{C}.$ According to the model, what temperature$\left(°C\right)$ will the water in the bottle eventually reach if the air temperature remains constant and the water is not drunk?

The number of customers, $n$, of a food shop $t$ months after it opens for the first time can be modelled as a continuous variable. It is suggested that the number of customers is increasing at a rate that is proportional to the square root of $n$. Form and solve a differential equation relating $n$ and $t$ to model this information.

The number of customers, $n$, of a food shop $t$ months after it opens for the first time can be modelled as a continuous variable. It is suggested that the number of customers is increasing at a rate that is proportional to the square root of $n$. Initially, $n=0$, and after $6$ months the food shop has $3600$ customers. Find how many complete months it takes for the number of customers to reach $6800$.(Round it off to single digit).

The number of customers, $n$, of a food shop $t$ months after it opens for the first time can be modelled as a continuous variable. It is suggested that the number of customers is increasing at a rate that is proportional to the square root of $n$. Initially, $n=0$, and after $6$ months the food shop has $3600$ customers.The food shop has a capacity of serving $14000$ customers per month. Show that the model predicts the shop will have reached its capacity sometime in the $12$ th month.

Doubling time is the length of time it takes for a quantity to double in size or value. The number of bacteria in a liquid culture can be modelled as a continuous variable and grows at a rate proportional to the number of bacteria present. Initially, there are $5000$ bacteria in the culture. After $3.5$ hours there are a million bacteria. What is the doubling time of the bacteria in the culture?(up to two decimals)

Doubling time is the length of time it takes for a quantity to double in size or value. The number of bacteria in a liquid culture can be modelled as a continuous variable and grows at a rate proportional to the number of bacteria present. Initially, there are $5000$ bacteria in the culture. After $3.5$ hours there are a million bacteria. What assumption does the model make about the growth of the bacteria and how realistic is this assumption?