# Max can borrow so that balance is decreasing

#### gGo

A loan repayment has an initial balance $$\displaystyle B_0$$ and monthly payments $400, where B(t) is the unpaid balance and B'(t) = 0.02B-400.​ What is the most you can borrow without going further into debt each month? Solution: We can calculate that $$\displaystyle B(t) = B_0e^{0.02t}+20000$$.​ I need to find B_0 so that B'(t) is not increasing, or $$\displaystyle B'(t) <= 0$$. $$\displaystyle B'(t) = 0.02\left(B_0e^{0.02t}+20000\right)-400 = 0.02 B_0e^{0.02t}$$​ But that last expression is always positive (assuming I borrowed a positive amount of money). #### AmandasMathHelp ##### New member Your mistake here is assuming that the constant in front of the e^.02t term is B_0, the initial amount. But if you plug in 0 right now you won't get an initial amount of B_0. So instead give that constant a variable and solve for it by setting the equation equal to B0 when t=0. gGo #### gGo ##### New member Your mistake here is assuming that the constant in front of the e^.02t term is B_0, the initial amount. But if you plug in 0 right now you won't get an initial amount of B_0. So instead give that constant a variable and solve for it by setting the equation equal to B0 when t=0. $$\displaystyle B(t) = (B_0 - 20000)e^{0.02t} + 20000 \\ B'(t) = 0.02(B_0 - 20000)e^{0.02t} = 0 \\ B_0 = 20000$$ #### AmandasMathHelp ##### New member You're equation for B(t) looks good now! What do you mean B_0 equals 2000? You mean for when there is debt? Because if you plug in 2000 into B'(t) you get 0.. not negative numbers. What is your condition for B'(t) to be negative? Hint: use an inequality like > or <. gGo #### gGo ##### New member You're equation for B(t) looks good now! What do you mean B_0 equals 2000? You mean for when there is debt? Because if you plug in 2000 into B'(t) you get 0.. not negative numbers. What is your condition for B'(t) to be negative? Hint: use an inequality like > or <. Ah yes, need to be clear. I was actually just recording the final work in case I ever come back here. But nonetheless, notice that if$B_0 < 20000\$, then $$\displaystyle B_0-20000 < 0$$ which would make

$$\displaystyle B'(t) = 0.02(B_0-20000)e^{0.02t} < 0$$​

for any value of $$\displaystyle t$$, thus the balance is decreasing.