Exponential functions > Real exponents
12345Real exponents

## Solutions to the exercises

Exercise 1
a

$A\left(10\right)=25000\cdot {1.1}^{10}\approx 64844$

b

$A\left(10\frac{7}{12}\right)\approx 68551$

c

$1.1$

d

${1.1}^{\frac{1}{12}}\approx 1.008$ so approximately $0.8$% per month.

e

$A\left(-5\right)\approx 15523$ en $A\left(-10\right)\approx 9639$

f

Check that $A\left(-5\right)\cdot {1.1}^{-5}=A\left(-10\right)$.

Exercise 2
a

1-1-2001: € 7518.15
1-1-2000: € 7092.60
1-1-1999: € 6691.13

b

On january 1, 1996.

c

He deposited € 5000 on $1$ January 1994.

Exercise 3
a

${g}_{\text{3 uur}}=\frac{3000}{1200}=2.5$

b

${g}_{\left(\text{1 hour}\right)}={\left(2.5\right)}^{\left(\frac{1}{3}\right)}\approx 1.357$ so 35.7% per hour.

c

$H\left(t\right)=1200\cdot {1.357}^{t}$

d

Around two and a quarter hours before $t=0$.

Exercise 4
a

0 - 1500: approximate yearly growth rate $1.00046$, so approximate growth percentage $0.05$% per year
1500 - 1800: approximate yearly growth rate $1.002313$, so approximate growth percentage $0.23$% per year
1800 - 1950: approximate yearly growth rate $1.00463$, so approximate growth percentage $0.46$% per year
1950 - 1986: approximate yearly growth rate $1.01944$, so approximate growth percentage $1.94$% per year

b

1500 - 1750: approximate yearly growth rate $1.00115$, so approximate growth percentage $0.12$% per jaar
1750 - 1800: approximate yearly growth rate $1.00814$, so approximate growth percentage $0.81$% per year
1986 - 1997: approximate yearly growth rate $1.01735$, so approximate growth percentage $1.74$% per year

Exercise 5

The permitted content is called $A$ , after the accident the actual content is $6A$ . Then ${\left(\frac{1}{2}\right)}^{t}\cdot 6A=A$ and this gives ${\left(\frac{1}{2}\right)}^{t}=\frac{1}{6}$ . Using the GRC you find $t\approx 2.58$ , so $2.58$ periods of $8$ days. That is $20.68$ days. The hay should be stored $21$ days.