Profitable zones Recall from chapters 3 and 4 that the profit P (æ) realized by selling x units of a given product is related to the cost C (x) to produce x units and the revenue R (x) by the formula P (x) = R (æ) – C (x). If we seek to maximize profit, which is the goal of any company in real world, the techniques of these chapters lead us to look at the critical points of P (x) ; in particular, if C(x) and R(x) are differentiable functions, we are interested in the solutions of the equation P' (x) = 0, which are also the solutions of the equation R' (x) = C' (x). In general, a graph of a typical cost and revenue functions - costs often initially exceed revenue, then fall below revenue as bulk manufacturing and transportation savings are realized, then eventually exceed revenue again as production capacity and market saturation points are reached. Proditabe e Companies try to always stay in a profitable zone. • Show that the profitable zone is bounded by two positive break-even points, where C(x) = R(x). · Explain using Calculus that the points (production levels x) of both maximum profit and maximum loss are indeed where C' (x) = R' (x). • Please elaborate how this confirms the strategy for maximizing profit that it is not enough to just find the critical points of P(x), but use the First Derivative Test, or the Second Derivative Test to determine the type of the extremum for P (æ) ( min or max) at the critical points of P (æ). To illustrate your understanding of the concept above, solve the problem below and share with your classmates the following: 1. The profitable zone of calendar production levels 2. The maximum profit and the level of production that will generate this maximum profit 3. The maximum loss and the level of production generating this maximum loss when the company is not in the profitable zone. Suppose that Best Custom Calendar Company models the coming year's revenue and cost functions, in thousands of dollars, to be R(x) = -x + 9x? and C (x) = 2x3 – 12x? + 30x , where x represents units of 1000 calendars and where the model is thought to be accurate up to approximately x = 5.5 . Round the dollar amounts to the nearest dollar.
Profitable zones Recall from chapters 3 and 4 that the profit P (æ) realized by selling x units of a given product is related to the cost C (x) to produce x units and the revenue R (x) by the formula P (x) = R (æ) – C (x). If we seek to maximize profit, which is the goal of any company in real world, the techniques of these chapters lead us to look at the critical points of P (x) ; in particular, if C(x) and R(x) are differentiable functions, we are interested in the solutions of the equation P' (x) = 0, which are also the solutions of the equation R' (x) = C' (x). In general, a graph of a typical cost and revenue functions - costs often initially exceed revenue, then fall below revenue as bulk manufacturing and transportation savings are realized, then eventually exceed revenue again as production capacity and market saturation points are reached. Proditabe e Companies try to always stay in a profitable zone. • Show that the profitable zone is bounded by two positive break-even points, where C(x) = R(x). · Explain using Calculus that the points (production levels x) of both maximum profit and maximum loss are indeed where C' (x) = R' (x). • Please elaborate how this confirms the strategy for maximizing profit that it is not enough to just find the critical points of P(x), but use the First Derivative Test, or the Second Derivative Test to determine the type of the extremum for P (æ) ( min or max) at the critical points of P (æ). To illustrate your understanding of the concept above, solve the problem below and share with your classmates the following: 1. The profitable zone of calendar production levels 2. The maximum profit and the level of production that will generate this maximum profit 3. The maximum loss and the level of production generating this maximum loss when the company is not in the profitable zone. Suppose that Best Custom Calendar Company models the coming year's revenue and cost functions, in thousands of dollars, to be R(x) = -x + 9x? and C (x) = 2x3 – 12x? + 30x , where x represents units of 1000 calendars and where the model is thought to be accurate up to approximately x = 5.5 . Round the dollar amounts to the nearest dollar.
Chapter1: Making Economics Decisions
Section: Chapter Questions
Problem 1QTC
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Transcribed Image Text:Profitable zones
Recall from chapters 3 and 4 that the profit P (x) realized by selling x units of a given product is related to the cost C (x) to produce x units and the revenue R (x) by the formula P (x)= R(x) – C (x). If we seek to
maximize profit, which is the goal of any company in real world, the techniques of these chapters lead us to look at the critical points of P (x); in particular, if C(x) and R(x) are differentiable functions, we are interested in the
solutions of the equation P' (x) = 0, which are also the solutions of the equation R' (x)= C' (x).
In general, a graph of a typical cost and revenue functions - costs often initially exceed revenue, then fall below revenue as bulk manufacturing and transportation savings are realized, then eventually exceed revenue again as
production capacity and market saturation points are reached.
Profitabe zone
Companies try to always stay in a profitable zone.
Show that the profitable zone is bounded by two positive break-even points, where C (x) = R ().
· Explain using Calculus that the points (production levels x) of both maximum profit and maximum loss are indeed where C' (x) = R' (x).
Please elaborate how this confirms the strategy for maximizing profit that it is not enough to just find the critical points of P(x), but use the First Derivative Test, or the Second Derivative Test to determine the type of the
extremum for P (x) (min or max) at the critical points of P (x) .
To illustrate your understanding of the concept above, solve the problem below and share with your classmates the following:
1. The profitable zone of calendar production levels
2. The maximum profit and the level of production that will generate this maximum profit
3. The maximum loss and the level of production generating this maximum loss when the company is not in the profitable zone.
Suppose that Best Custom Calendar Company models the coming year's revenue and cost functions, in thousands of dollars, to be R (x) =
-x3 + 9x? and C (x) = 2x3
12x2 + 30x , where x represents units of 1000
calendars and where the model is thought to be accurate up to approximately x = 5.5 . Round the dollar amounts to the nearest dollar.
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