Problem 99 - Calculus Early Transcendentals

Question

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Step-by-Step Solution

Answer

#Answer# The profit for x = 20 and y = 10 is positive (z = 50). There are several integer pairs (x, y) that would make the company break even, such as (0, 40), (20, 0), and (10, 20). The break-even point is a line that represents all combinations of x and y values that result in a profit of zero.

Step 1: Graph the profit plane

The equation for the profit is given as \(z=10x+5y-200\). To graph this equation in the range of \([0,40]\times[0,40]\times[-400,400]\), we need to create a 3D plot with axis labels representing the number of long-sleeved T-shirts (x-axis), the number of short-sleeved T-shirts (y-axis), and the profit (z-axis). Unfortunately, we cannot draw a direct representation of this graph in this text, but you can plot it using graphing software or online tools like Desmos or WolframAlpha.

Step 2: Determine profit for given values of x and y

Now, we have to determine whether the profit is positive or negative for \(x=20\) and \(y=10\). Plugging these values into the profit equation, we get: \(z = 10(20) + 5(10) - 200\) \(z = 200 + 50 - 200\) \(z = 50\) The profit is positive, as \(z=50 > 0\).

Step 3: Determine break-even point

To determine the values of \(x\) and \(y\) that would make the company break even, we need to find the point where the profit, \(z\), is zero. So, we need to solve the equation \(z = 10x + 5y - 200 = 0\) for \(x\) and \(y\). Rearranging the equation gives us \(10x + 5y = 200\). We can simplify it to \(2x + y = 40\). There are several possible integer pairs \((x, y)\) that satisfy this equation, such as \((0, 40)\), \((20, 0)\), and \((10, 20)\). But the break-even point isn't just a single point on the graph. It's a line that represents all combinations of \(x\) and \(y\) values that make the company break even. You can mark this line on your graph along with the values of \(x\) and \(y\) that keep the profit at zero.