2022 VJC J1 MYE Q10
A shrimp farmer cultivates marine shrimps in tanks. There are $4000$ marine shrimps in a tank at the start of January $2022$. At the start of each month, the farmer will harvest $500$ marine shrimps for sale. At the end of each month, the number of marine shrimps in the tank will increase by $20%$ from the number at the start of the month after the shrimps are harvested for sale.
(a)
(i) Find the number of marine shrimps in the tank at the end of February $2022$.
[2]
(ii) Show that the number of marine shrimps in the tank at the end of $n$ months after the start of January $2022$ is $1000{{\left( 1.2 \right)}^{n}}+3000$.
[4]
(iii) The tank will be too congested for marine shrimps to grow well when there are more than $8000$ marine shrimps in the tank. Find when will the tank first become too congested for the marine shrimps to grow well.
[2]
(b)
Salt water is added to the tank at the start of each month to ensure that the marine shrimps are growing well. In January $2022$, $25$m$^{3}$ of salt water is added to the tank. In each subsequent month, the amount of salt water added to the tank is $r%$ of the amount of salt water added in the previous month.
(i) Assuming $r=55$, find the amount of salt water added to the tank on September $2022$.
[2]
(ii) The capacity of the tank is $60$m$^{3}$. Find the range of values of $r$ such that the theoretical maximum total amount of salt water added to the tank will never exceed the capacity of the tank.
[3]
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