Calculus problem example

I'll be taking pre-calc soon, and I've been wondering about something.

I know that some problems can be solved with algebra.

Example:

Tickets for a school play are $12 for adults and $8 for students. The school sold 200 tickets for Friday night. If the school earned $1,860 for the show, how many $12 tickets did they sell.

I can solve this example writing

12x + 8y = 1860
x + y = 200
(Can solve by elimination or substitution)

So, I was wondering if anyone could give me an example of a problem that cannot be solved by using only Algebra and must us Calculus to solve it.

How about optimization problems such as this

The first problem in BAdhi's link can be solved without calculus. By AM-GM inequality, . Equality occurs if and only if x = 2y (x = 250, y = 125).


Other optimization problems don't have easy algebraic/geometric solutions and therefore must be solved using calculus. For example, find the area bounded by the x-axis and , .