Investment problem with strange answer.

Question

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.

Answer 1

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

Answer 2

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
• ...