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In this lesson, we will learn how to solve real-world problems on exponential growth, such as population growth and compounded interest.

Q1:

When caffeine is metabolized by our body (that is, when our body breaks down, uses, and absorbs caffeine), the decreasing quantity of caffeine can be modelled by the following function π = π π 0 β 0 . 1 π‘ , where π‘ is the number of hours after an intake of π 0 . What is the half-life of caffeine in our body (that is, how long does it take for our body to break down half of the caffeine)? Round your answer to the nearest hour.

Q2:

The function π ( π‘ ) = π΄ π π‘ represents a population, in millions, π‘ years after 1970 that is growing at an annual rate of 3 . 5 % and started at 13.2 million in 1970. What is the value of π ?

Q3:

The number of cars worldwide, π‘ years after 2015, can be modeled by the formula π = 1 0 β π ο― ο¦ ο ο¦ ο¨ ο ο ο . In what year will there be 1.4 billion cars worldwide?

Q4:

There is a gap of 3 mm between the floor and one of the legs of a table. How many times would a sheet of paper of thickness 0.08 mm need to be folded to fill that gap?

Q5:

The number of users of a new search engine is increasing every month and can be found using the equation π¦ = 5 0 0 ( 1 . 1 9 ) π₯ , where π¦ represents the number of users and π₯ represents the number of months since the search engineβs launch. If the search engine was launched on the 1st of March, in which month would the search engine have 2 0 0 0 users?

Q6:

The number of tourists visiting a theme park increases every year and can be found using the equation π¦ = 1 . 1 ( 1 . 0 4 5 ) ο , where π¦ million is the number of visitors π‘ years after 2010. If the number of visitors continues to increase at the same rate, in what year will the park first reach 2 million visitors?

Q7:

The value of a car depreciates by 1 3 % every year. If a car cost $ 7 5 0 0 0 when bought new, how old would it be when its value has dropped to $ 4 0 0 0 0 ? Give your answer in years to two decimal places.

Q8:

The value of a car depreciates at a rate of 1 5 % each year.

Write an equation that can be used to calculate π , the value of a car, in dollars, π‘ years after it was purchased for πΆ d o l l a r s .

What is the total depreciation in the carβs value over 6 years? Give your answer to the nearest percent.

Q9:

A population of fruit flies quadruples every 3 days.

Write an equation that could be used to calculate π , the number of fruit flies after π‘ days, if the initial population was 400.

How many fruit flies will there be after 5 days?

Q10:

In the US, the proportion of waste that was recycled has roughly tripled between 1985 and 2005. Using an exponential model for this proportion, find in which year the proportion of waste had roughly doubled with respect to the value in 1985.

Q11:

In a laboratory, a bacteria population quadruples every hour. The population was first measured to be 50 bacteria. Write an equation that can be used to find π΅ , the bacteria population after π‘ hours.

Q12:

Rewrite π ( π‘ ) = 3 . 6 2 ( 1 . 0 2 9 ) π‘ in the form π ( π‘ ) = π ( 2 ) 0 π‘ π , with π to two decimal places. What is the significance of the number π ?

Q13:

A scientist is considering two termite species: π΄ and π΅ . At the start of the experiment, there are 1 2 3 3 of π΄ and 1 6 4 0 of π΅ . They both increase exponentially: the smaller group π΄ at 1 . 3 % per day, which is higher than π΅ βs 0 . 4 % . The scientist believes that, despite the fact that π΅ had a head start, π΄ will eventually surpass π΅ in terms of population given its higher rate. She also believes that this will happen by day 30. Is her estimate correct? To use the model, you must round to the nearest integer.

Q14:

A population grew from 3.62 million to 4.604 million in ten years. What is the annual percentage growth rate of this population? Give your answer to 2 decimal places.

Q15:

A population that obeys an exponential growth rate of 2 . 4 3 % per year grows by a fixed percentage every two years. What is this percentage?

In what period will this population grow by 50% approximated to one decimal place?

Q16:

The number, in millions, of trucks on the road worldwide in the year π¦ can be modeled by π = 3 7 7 Γ 1 . 0 3 ( ο ο± ο¨ ο ο¦ ο§ ο« ) . In what year does the model predict that there would be 450 million trucks worldwide?

Q17:

The population of Malawi is modeled to be 3 . 6 2 ( 1 . 0 2 9 ) π‘ million at the beginning of the π‘ th year after 1960. During what year will the population reach 20 million?

Q18:

On July 5, green algae was found on the bottom of a swimming pool whose width is 6 m and length is 12 m. If the area, in mm^{2}, the algae covers π‘ days later is given by π΄ = 4 . 3 β 2 π‘ 3 , when will the algae completely cover the bottom of the swimming pool?

Q19:

A population of seagulls grows from 75 to 102 in 6 months. Find the continuous growth rate. Give your answer as a percentage to one significant figure.

Q20:

The population of bacteria found in raw milk cheeses was found to increase by a factor of 10 after 10 hours at a temperature of 3 3 β C .

If the population started at 50 bacteria, how long would it take it to reach 300 bacteria, assuming exponential growth? Give the answer in hours and minutes.

How long would it take for the number of bacteria to double?

Q21:

A certain growth of cancerous cells increased from 400 initially to 440 one month later. How many months will it take for the initial number of cancer cells to double? If necessary, round your answer to one decimal place.

Q22:

Fruit fly populations can grow exponentially.

The function π ( π‘ ) = π π ο¦ ο can be used to model a fruit fly population, where π ο¦ is the initial population, and π‘ is the time in days after the population was first measured.

If a population of fruit flies triples every 2 days, find the value of π .

If the function is rewritten in the form π ( π€ ) = π π ο¦ ο , where π€ is the time in weeks, what is the value of π ?

Q23:

The population of rabbits on a farm grows exponentially. If there are currently 245 rabbits and the growth rate is 2 3 % , find a function π ( π‘ ) to describe the number of rabbits after π‘ years.

Q24:

Shadyβs bank account gives him 5 . 6 % interest on his balance each month. He models his balance after π β₯ 1 months with the recursive formula π = ( 1 + 0 . 0 5 6 ) π π π β 1 . If his initial deposit is 450.00, after how many months will his balance be greater than $600?

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