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

i. C

ii. C

iii. A

iv. B

Step-by-step 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<0 \end{gathered}`

For the given question;

i.

`y=(0.9)^{(t)/(2)}`

the values of r is;

`\begin{gathered} r=0.9 \\ 0.9<1 \\ r<1 \end{gathered}`

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<1 \\ r<1 \end{gathered}`

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<1 \\ r<1 \end{gathered}`

it is an exponential decay.

The rate of decay is;

`\begin{gathered} (1-r)*100\text{\%} \\ =(1-0.85)*100\text{\%} \\ =15\text{\%} \end{gathered}`

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

B.

v.

`y=((1)/(2))^t`

The value of r is;

`\begin{gathered} r=(1)/(2)=0.5 \\ 0.5<1 \\ r<1 \end{gathered}`

It is an exponential decay.

The rate of decay is;

`\begin{gathered} (1-r)*100\text{\%} \\ =(1-0.5)*100\text{\%} \\ =50\text{\%} \end{gathered}`

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

D.