Question

prompt
suppose that you want to start a new business. you know that the revenue is a function of the number of widgets that your new company sells and that the cost is a function of the number of widgets that your new company produces. given this information, what method would you use to solve for the break - even point? explain what your answer means
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Explanation:

1. Define Functions: Let the number of widgets be \(x\). The revenue function \(R(x)\) models the total money earned from selling \(x\) widgets (e.g., if each widget is sold at a price \(p\), \(R(x)=p\times x\) for a linear revenue model). The cost function \(C(x)\) models the total money spent to produce \(x\) widgets (it can include fixed costs like rent and variable costs like the cost of raw materials per widget, e.g., \(C(x)=FC + vc\times x\), where \(FC\) is fixed cost and \(vc\) is variable cost per widget).
2. Set Functions Equal: At the break - even point, the total revenue earned is equal to the total cost incurred. So we set up the equation \(R(x)=C(x)\).
3. Solve for \(x\): By solving the equation \(R(x) = C(x)\) for \(x\), we find the number of widgets that need to be produced and sold so that the company neither makes a profit nor incurs a loss. For example, if \(R(x)=10x\) (selling 10 per widget) and \(C(x)=5x + 100\) (fixed cost of 100 and 5 per widget to produce), we solve \(10x=5x + 100\). Subtract \(5x\) from both sides: \(10x-5x=5x + 100 - 5x\), which gives \(5x = 100\). Then divide both sides by 5: \(x=\frac{100}{5}=20\). So the break - even point is at \(x = 20\) widgets.

Answer:

To find the break - even point, we set the revenue function equal to the cost function and solve for the number of widgets (\(x\)). Let \(R(x)\) be the revenue function and \(C(x)\) be the cost function. The break - even point occurs when \(R(x)=C(x)\).