A tank with a capacity of 1000 L is full of a mixture of water and chlorine with a concentration of 0.02 g of chlorine per liter
1 answer:
At the start, the tank contains
(0.02 g/L) * (1000 L) = 20 g
of chlorine. Let <em>c</em> (<em>t</em> ) denote the amount of chlorine (in grams) in the tank at time <em>t </em>.
Pure water is pumped into the tank, so no chlorine is flowing into it, but is flowing out at a rate of
(<em>c</em> (<em>t</em> )/(1000 + (10 - 25)<em>t</em> ) g/L) * (25 L/s) = 5<em>c</em> (<em>t</em> ) /(200 - 3<em>t</em> ) g/s
In case it's unclear why this is the case:
The amount of liquid in the tank at the start is 1000 L. If water is pumped in at a rate of 10 L/s, then after <em>t</em> s there will be (1000 + 10<em>t</em> ) L of liquid in the tank. But we're also removing 25 L from the tank per second, so there is a net "gain" of 10 - 25 = -15 L of liquid each second. So the volume of liquid in the tank at time <em>t</em> is (1000 - 15<em>t </em>) L. Then the concentration of chlorine per unit volume is <em>c</em> (<em>t</em> ) divided by this volume.
So the amount of chlorine in the tank changes according to
which is a linear equation. Move the non-derivative term to the left, then multiply both sides by the integrating factor 1/(200 - 5<em>t</em> )^(5/3), then integrate both sides to solve for <em>c</em> (<em>t</em> ):
There are 20 g of chlorine at the start, so <em>c</em> (0) = 20. Use this to solve for <em>C</em> :
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Answer:
No lower than $208, and no higher than $956.8
Step-by-step explanation:
Notice that the revenue (income) function is represented by a quadratic expression (a parabola with branches pointing down), and the cost function is linear with negative slope.
When we plot them, we obtain the graphs shown in the attached image. Notice as well that there are two points of intersection for these two functions. The revenue exceeds (lays above) the cost in between the two points of intersection (marked in red and green).
For the company to make a profit, the revenue has to be larger than the cost, so we study the limiting values for that to happen, which is the points of intersection of these two curves.
The first one (pictured in red) is associated with coordinates (35, 33488) which tells us that is the production of 35 chairs with a revenue of $33,488. That is a cost per chair of $33,488 / 35 = $956.8
The second intersection (pictured in green) is associated with coordinates (125, 26000), which tells us that is the production of 125 chairs with a revenue of $26,000. That is a cost per chair of $26000 / 125 = $ 208
So these values give us the minimum and maximum that the company can charge to make a profit.
