MEDIUM
AS and A Level
IMPORTANT
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A ball in the shape of a sphere is being filled with air. After t seconds, the radius of the ball is r cm. The rate of increase of the radius is inversely proportional to the square root of its radius. It is known that when t=4 the radius is increasing at the rate of 1.4 cm s-1 and r=7.84. How much air was in the ball at the start? (in cm3 round up to one decimal)

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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 8

Maria makes a large dish of curry on Monday, ready for her family to eat on Tuesday. She needs to put the dish of curry in the refrigerator but must let it cool to room temperature, 18°C, first. The temperature of the curry is 94°C when it has finished cooking.
At 6pm, Maria places the hot dish in a sink full of cold water and keeps the water at a constant temperature of 7°C by running the cold tap and stirring the dish from time to time. After 10 minutes, the temperature of the curry is 54°C.
It is given that the rate at which the curry cools down is proportional to the difference between the temperature of the curry and the temperature of the water in the sink.
At what time can Maria put her dish of curry in the refrigerator?

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 9

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?

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.