In business, we use the equation $$P(u) = R(u) - C(u)$$ where $P(u)$ is the profit you make from selling $u$ tons of some product, after you pay expenses. The function $R(u)$ represents the revenue from selling the product, and $C(u)$ represents the cost you must pay to manufacture the product. The graph below shows the cost and revenue functions, where the input, $u\text{,}$ is the number of tons of the product sold. The vertical intercept of the Cost function is at $(0, 100)\text{.}$ Click and drag this point along the Cost curve $y = C(u)\text{.}$ This will move points on the graphs of the Revenue and Profit functions. Notice that the height of the point $P(u)$ is equal to the difference in the heights of points on the $R(u)$ and $C(u)$ graphs.

1. Move the point along the graph of $C(u)$ until the profit is zero. At approximately how many tons will the profit be zero (give your answer to the nearest whole number)? Note: This value is called the break even point, after which the business begins to make actual profit.
   tons
2. Now click the box to show the graph of $P(u)\text{.}$ Notice that it begins with negative outputs, increasing to the break even point, and becoming positive thereafter.
   • If $C($6$) =$221.2 and $R($6$) =$240, then what is $P($6$)\text{?}$
     Answer:
   • If it costs 124.2 thousand dollars to produce 1 tons, and the revenue is 40 thousand dollars, then what is $P($1$)\text{?}$
     Answer: