Algebra, STD2 2026 HSC 25

A large rechargeable battery has a maximum storage capacity of 40 kWh when new.

After 1 year, the maximum storage capacity is 32 kWh.

The process of decay can be modelled by the function

$y=ka^t$

where $y$ is the maximum storage capacity of the battery, in kWh, $t$ years after it starts being used.

What are the values of $k$ and $a$? (2 marks)
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Interpret the value of $a$ in this context. (1 mark)
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Show Answers Only

a. $k=40,\quad a=0.8$

b. $\text{Since $a=0.8$, maximum storage capacity each year is equal to 80%}$

$\text{of the maximum capacity of the previous year.}$

$\text{(i.e. maximum storage capacity decreases by 20% each year)}$

Show Worked Solution

a. $\text{At}\ \ t=0, \ y=40:$

$40=ka^0\ \ \Rightarrow \ \ k=40$

$\text{Find $a$ given at}\ \ t=1, \ y=32:$

$32=40a^{1}\ \ \Rightarrow\ \ a=\dfrac{32}{40}=0.8$

b. $\text{Since $a=0.8$, maximum storage capacity each year is equal to 80%}$

$\text{of the maximum capacity of the previous year.}$

$\text{(i.e. maximum storage capacity decreases by 20% each year)}$