Answer and Explanation:
Gina wanted to swim at the pool. She is allowed to spend at most $30. The cost to swim in the pool is 3 dollars per hour plus a flat fee of $3. How many hours can Gina swim without going over her spending limit?
This problem can be represented by .
'x' would be the number of hours. "At most" means less than or equal to. So, the value of x would have to be less than or equal to 30.
So, Gina would have a maximum of 9 hours to swim in the pool.
Hope this helps.
1. I suppose the ODE is supposed to be
Solving for gives
which is undefined when . The interval of validity depends on what your initial value is. In this case, it's
, so the largest interval on which a solution can exist is
.
2. Separating the variables gives
Integrate both sides. On the left, we have
where we substituted - or
- and
- or
.
On the right, we have
So
I'll leave the solution in this form for now to make solving for easier. Given that
, we get
and so our solution is