2025 EJC Promo Q13

The effective range of an electric car refers to the distance that an electric car can travel on a single full charge. Electrica, an electric car manufacturer, releases the Model LX with an initial effective range of 460 km to celebrate its \(10^{\text{th}}\) anniversary.

The effective range decreases over time due to battery degradation. In the first year of ownership, Model LX’s effective range decreases by 5.6 km. In each subsequent year, the decrease in effective range is 0.75 km more than the decrease in the previous year. For instance, the effective range decreases further by 6.35 km in the 2nd year.

1. Show that the decrease in effective range in the \(7^{\text{th}}\) year is 10.1 km.[2]
2. A battery replacement is recommended once the effective range falls below 75% of the initial range. Determine the year in ownership when a battery replacement is first recommended.[4]

On 1 January 2025, Ben purchased a brand-new Model LX and took up a car loan of $150 000 from a bank. The loan is such that an interest of 0.2% is added to the outstanding loan amount at the start of every month, with the first interest amount added on 1 January 2025. Ben makes a monthly repayment of $x on the \(12^{\text{th}}\) of every month, with the first repayment on 12 January 2025.

4. Show that the outstanding loan amount, in dollars, at the end of \(n\) months is[3]
   \(150\ 000(1.002^n)-500x(1.002^n-1).\)
5. 1. Suppose Ben made the last monthly repayment of \(x on 12 Dec 2029 to fully pay off the loan. Find the value of \)x to the nearest cent.[2]
   2. Use your answer in part d(i) to calculate the total interest paid on the loan.[1]