In $2022,$ the average annual cost of groceries for a family living in Canada was $$15,532.$ The cost of groceries increased by $$1,060$ for a family from $2022$ to $2023,$ and this annual increase in cost is predicted to continue to increase by $$255$ each subsequent year.

Assuming this trend continues indefinitely, use anti$-$derivatives to express the cost of groceries for a family as a function of $t,$ where $t$ represents the number of years that have passed since $2022.$Please explain all steps and explain