MEDIUM
AS and A Level
IMPORTANT
Earn 100

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 m0. What fraction of the original amount of carbon-14 would be present in this sample after 2500 years?

Important Questions on Differential Equations

MEDIUM
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 10
Anya carried out an experiment and discovered that the rate of growth of her hair was constant. At the start of her experiment, her hair was 20 cm long. After 20 weeks, her hair was 26 cm long. Form and solve a differential equation to find a direct relationship between time, t, and the length of Anya's hair, L.
MEDIUM
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 11
A bottle of water is taken out of a refrigerator. The temperature of the water in the bottle is 4°C. The bottle of water is taken outside to drink. The air temperature outside is constant at 24°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°C. How long does it take for the water to warm 20°C?
MEDIUM
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 11
A bottle of water is taken out of a refrigerator. The temperature of the water in the bottle is 4°C. The bottle of water is taken outside to drink. The air temperature outside is constant at 24°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°C. According to the model, what temperature°C will the water in the bottle eventually reach if the air temperature remains constant and the water is not drunk?
EASY
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 12
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.
EASY
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 12
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).
EASY
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 12
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.
MEDIUM
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 13
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)
MEDIUM
AS and A Level
IMPORTANT
Cambridge International AS & A Level Mathematics : Pure Mathematics 2 & 3 Course Book>Chapter 10 - Differential Equations>EXERCISE 10B>Q 13
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?