Exponential functions > Real exponents
12345Real exponents

Exercises

Exercise 1

A suburb of a big city grows according to the formula $A\left(t\right)=25000\cdot 1.{1}^{t}$ for a number of years. $A\left(t\right)$ is the number of inhabitants at time $t$, where $t$ is the time in years and $t=0$ on 1-1-1995. Assume the suburb will continue to grow according to this formula.

a

How many inhabitants does the suburb have on 1-1-2005?

b

How many inhabitants does the suburb have on 1-8-2005?

c

What is the growth rate per year?

d

What is growth rate per month?

e

Calculate the number of inhabitants on $1$ January in the years 1990 and 1985.

f

Show that ${\left(1.{1}^{\left(-5\right)}\right)}^{2}=1.{1}^{\left(-10\right)}$. Use the years 1995, 1990 and 1985.

Exercise 2

On $1$ January 2002 somebody has a balance in his savings account of € 7969.24. The capital has received a yearly interest of 6%.

a

Calculate the balance on $1$ January 2001, $1$ January 2000 and $1$ January 1999.

b

In what year will the balance be $7969.24\cdot 1.{06}^{\left(-6\right)}$ ?

c

The savings account holder probably deposited a round amount when he started saving. When did he start, and with what amount?

Exercise 3

A bacterial colony grows exponentially. Its number has grown from $1200$ to $3000$ in three hours.

a

What is the growth rate per $3$ hours?

b

Calculate the growth rate per hour.

c

What formula can you construct to describe the growth of this colony if $H\left(t\right)$ is the number of bacteria and $t$ the time in hours? Take $t=0$ at the moment that there are $1200$ bacteria.

d

At what moment were there only $600$ bacteria?

Exercise 4

Since the beginning of the calendar the world population has grown faster and faster. The world population of $300$ million at the beginning of the calendar doubled in fifteen hundred years. In 1750 there were $800$ million people and only fifty years later $1.2$ billion. No more than $150$ years later the world population had doubled again (to $2.4$ billion in 1950). In 1986 the world population was $4.8$ billion. In 1997 there were $1$ billion people more than in 1986. In 2000 there were $6$ billion people and in 2050 Earth may well be home to $9$ billion people.

a

The text refers to several different periods.
Calculate the yearly growth rate for those periods in which the world population doubled.

b

Calculate the yearly growth rate for the other periods as well.

Exercise 5

The radioactive substance iodine-131 is generated in a nuclear explosion. Because the fall-out drops onto the grass, the iodine-131 content of the hay becomes too high. Milk from cows fed with this hay is no longer suited for human consumption. After an accident in a nuclear plant the iodine-131 content of the hay in the surroundings of the plant is six times the allowed content. The half-life of iodine-131 is eight days.

How many days should the hay be stored before it can be fed to cows again?