Question

I need some help with this math question I am supposed to match each one of them with the following solutions attached in the second image

Answer 1

Answer:

i. C

ii. C

iii. A

iv. B

Explanation:

Exponential growth / decay can be shown using the formula below;

$y=a(r)^{kt}$

Where;

$\begin{gathered} y=\text{ final amount} \\ a=\text{ initial amount} \\ r=\text{exponential decay or growth.} \\ t=\text{time} \\ k=\text{constant} \end{gathered}$

The difference between exponential decay and growth is;

$\begin{gathered} \text{ For exponential growth;} \\ r>0 \\ \text{ For exponential decay;} \\ r$

For the given question;

i.

$y=(0.9)^{\frac{t}{2}}$

the values of r is;

$\begin{gathered} r=0.9 \\ 0.9$

Therefore, the function reveals exponential decay

C.

ii.

$y=(0.81)^{6t}$

The same rule as in i applies here;

the values of r is;

$\begin{gathered} r=0.81 \\ 0.81$

Therefore, the function reveals exponential decay

C.

iii.

$y=(1.08)^{t+6}$

the values of r is;

$\begin{gathered} r=1.08 \\ 1.08>1 \\ r>1 \end{gathered}$

Therefore, the function reveals exponential growth

A.

iv.

$y=(0.85)^t$

the values of r is;

$\begin{gathered} r=0.85 \\ 0.85$

it is an exponential decay.

The rate of decay is;

$\begin{gathered} (1-r)\times100\text{\%} \\ =(1-0.85)\times100\text{\%} \\ =15\text{\%} \end{gathered}$

Therefore, the rate of decay of the function is 15%.

B.

v.

$y=(\frac{1}{2})^t$

The value of r is;

$\begin{gathered} r=\frac{1}{2}=0.5 \\ 0.5$

It is an exponential decay.

The rate of decay is;

$\begin{gathered} (1-r)\times100\text{\%} \\ =(1-0.5)\times100\text{\%} \\ =50\text{\%} \end{gathered}$

Therefore, the rate of decay of the function is 50%.

D.