How much of Lemonade to sell?

Let the selling price of a glass of lemonade be $p$ and the number of glasses sold be $q$ . Then $q = 197 - 100(p-1) = 297 - 100p$ . The the income in a day is:

$\begin{aligned} I & = pq \\ & = 297p - 100p^2 \\ & = 1.485^2\times 100 - 100\blue{(p - 1.485)^2} & \small \blue{\text{Since }(p-1.485)^2 \ge 0} \\ \implies I & \le 220.5225 \end{aligned}$

Since you can only price the lemonade to the nearest 1 cent. The highest income can either be $\begin{cases} I (1.48) = 297(1.48) - 100(1.48^2) = 220.52 \\ I (1.49) = 297(1.49) - 100(1.49^2) = 220.52 \end{cases}$ . Therefore the highest income in a day is $\$\boxed{220.52}$ .

Let $x$ denote the number of lemonade glasses sold sold per day. Then the number of glasses not sold is $197-x$ . That means the increase in cost per glass of lemonade is $\frac{197-x}{100}$ . So the cost of one glass of lemonade is $1 + \frac{197-x}{100}$ . Let $C(x)$ denote the income that can be earned by selling $x$ lemonade glasses. Then

$C(x) = \bigg(1 + \large\frac{197-x}{100}\bigg)x$

$\Rightarrow C(x) = \large\frac{(297-x)x}{100}$

This is a quadratic polynomial with downward parabola (since the coefficient of $x^2$ is negative ) whose maximum occurs at the midpoint of the zeroes of this polynomial. Zeroes of this polynomial are $0$ and $297$ . Let $x*$ denote the number of lemonade glasses sold per day for maximum income. Then

$\hspace{13pt}x* = \lfloor 148.5\rfloor$ or $\lceil 148.5\rceil\newline \Rightarrow x* = 148$ or $149$

$C(x*) = \large\frac{(297-148)\cdot 148}{100} = \frac{149\cdot 148}{100} = \small 220.52$

Hence the maximum income that can be earned is $\$\,220.52$

The total income can be expressed as the following function I(x) = (197-x)*(1 + x/100) where x is the number of glasses not sold per day.

I(x) = 197 + 0.97 * x - (x^2/100)

Maximum of this function will occur when dI(x)/dx = 0 and d^2I(x)/dx < 0 , Solving for dI(x)/dx = 0 yields the equation 0.97 - x/50 = 0, solving for x yields x = 48.5 as x has to be integer , inspecting I(x) at x = 48 and at x = 49 results in a value of 220.52 for the total income.

The second derivative is always negative and so the maximum Income that can be earnt is by selling 148 or 149 glasses at $1.49 or $1.48. The maximum income is hence $220.52