Maximum points in a rowing and planking contest

Question

I am trying to solve a problem that was asked by a friend of mine who is attending high school. As it has something to do with linear optimization, I think I should be more than capable of solving the problem as I got really good results from my recent course in linear optimization. However, I am struggling to solve the problem. The problem is as follows:

Suppose you are competing with another class at school about getting the highest amount of points by either rowing or planking.

The rules are:

1. Four people have to row at once and four people have to do the plank at once.
2. You have to row for six minutes and do the plank for three minutes.
3. The class may only use 30 minutes total or less
4. The class may have 28 pupils and may only do one activity at once
5. The class gets 5 points for rowing once and 3 points for planking once.

So basically we must have a maximization problem

$$\max_{x,y} 5x + 3y$$

with the following constraints

\begin{align} 6x + 3y & \leq 30 \\ x & \leq 8 \\ y & \leq 4 \\ x, y & \geq 1 \end{align}

but what about the first rule. How would one incorporate such a thing? And does it even matter if they at $28$ pupils? Wouldn't just $4$ be enough?

Answer 1

Additional constraint could be that each pupil does just one activity, otherwise there would be the case that there are only 4 people required throughout as you pointed out. In this case, the problem formulation I see is as follows (slightly different from yours)

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Let $x$ be the number of times the class rows, and $y$ be the number of times they plank.

Maximize: \begin{align*} \max_{x,y} 5x + 3y \end{align*} subject to \begin{align*} 6x + 3y &\leq 30 \\ 4 x + 4y &\leq 28 \\ x,y&\geq 0 \\ \end{align*}