# Investment problem with strange answer.

I tried solve that problem and for me the maximum income is $$3000$$ because $$1000\times 3$$.

But the answer is $$3600$$! Can anybody explain me how it works? Thanks

James has $$\1000$$ and wants to invest it in a project. He knows that each dollar brings $$\2$$ income per month. He may rent a number of billboards for $$\100$$ each. Each billboard increases one dollar’s income by $$\1$$ per month. Find the James’ maximal total month income. Write your answer in dollars.

With no billboards, James gets $$\2$$ for every dollar. Let $$x$$ be the number of billboards bought. Then, for each dollar James would get $$\(x+2)$$, and the amount of money left is $$1000-100x$$. You can set the income up as a function: $$I(x) =(1000-100x)(x+2) =100(10-x)(x+2)$$ This is a down - parabola with roots $$-2$$ and $$10$$, so the maximum would occur at $$\frac{10+2}{2}-2=4$$. So, $$I_{\text{max}}=I(4) =3600$$

• Ok. Thank you but I don't understand yet, why he couldn't buy 10 billboards and increase for all money, and his income will be $3$ for one dollar – Krutya May 22 '20 at 18:48
• @Krutya Because $10$ billboards would cost him \$1000 and he would be left with no money, hence no income. – Tavish May 22 '20 at 18:51
• OMG understood;-) Thank you amigo! – Krutya May 22 '20 at 18:52
• @Krutya You’re welcome. – Tavish May 22 '20 at 18:53

I think the interpretation is that James can buy any number of billboards for $$\100$$ each; if he buys $$k$$ billboards he gets $$\(2+k)$$ of income for every dollar he invests (his investment is not included in the income).

So his options are:

• buy no billboards, invest $$\1000$$, get $$\2000$$ income
• buy $$1$$ billboard, invest $$\900$$, get $$\2700$$ income
• buy $$2$$ billboards, invest $$\800$$, get $$\3200$$ income
• buy $$3$$ billboards, invest $$\700$$, get $$\3500$$ income
• buy $$4$$ billboards, invest $$\600$$, get $$\3600$$ income
• buy $$5$$ billboards, invest $$\500$$, get $$\3500$$ income
• ...