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, $$I_{\text{max}}=I(4) =3600$$

  • $\begingroup$ 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 $\endgroup$ – Krutya May 22 '20 at 18:48
  • 1
    $\begingroup$ @Krutya Because $10$ billboards would cost him $1000 and he would be left with no money, hence no income. $\endgroup$ – Tavish May 22 '20 at 18:51
  • $\begingroup$ OMG understood;-) Thank you amigo! $\endgroup$ – Krutya May 22 '20 at 18:52
  • $\begingroup$ @Krutya You’re welcome. $\endgroup$ – 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
  • ...

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.