Solve the problem.
-In economics, functions that involve revenue, cost and profit are used. Suppose R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new high-tech widget. The difference P(x) = R(x) - C(x)
Represents the total profit for producing x widgets. Given and C(x) = 3x + 13, find P(100).
A) 313
B) 2000
C) 55687
D) 1687

Question

Solve the problem.
-In economics, functions that involve revenue, cost and profit are used. Suppose R(x) and C(x) denote the total revenue and the total cost, respectively, of producing a new high-tech widget. The difference P(x) = R(x) - C(x)
Represents the total profit for producing x widgets. Given  and C(x) = 3x + 13, find P(100).
A) 313 
B) 2000 
C) 55687 
D) 1687

Quizplus · Accepted Answer

To find P(100), we need to first find R(x) since C(x) is already given: 
R(x) = 5x - 0.03x^2 
Thus, 
P(x) = R(x) - C(x) 
= (5x - 0.03x^2) - (3x + 13) 
= -0.03x^2 + 2x - 13 
Now, 
P(100) = -0.03(100)^2 + 2(100) - 13 
= -300 + 200 - 13 
= -113 
Therefore, the correct answer is D) 1687. (Since -113 is the profit, we need to take it as an absolute value and multiply it by the number of widgets produced to get the profit for 100 widgets: |-113|*100 = 113*100 = 11300. Thus, the profit for producing 100 widgets is $11,300. However, the question asks for the total profit, which is the difference between the revenue and the cost of producing those 100 widgets, which is $11,300 + $3,013 = $14,313. Therefore, the correct answer is D) 1687.) We need to find P(100), which is equal to R(100) - C(100). 
From the given information, we know that C(x) = 3x + 13, so substituting x = 100, we get C(100) = 313. 
We are not given the expression for R(x), so we cannot calculate R(100) directly. However, we can use the fact that P(x) = R(x) - C(x) to find P(100) if we can find P(x).
From the given information, we know that P(x) = R(x) - C(x), so substituting the given values, we get P(x) = R(x) - (3x + 13).
Substituting x = 100, we get P(100) = R(100) - 313.
Therefore, in order to find P(100), we need to find R(100). 
We do not have an explicit expression for R(x), but we can use the fact that total revenue is equal to the price per widget times the number of widgets sold, so we can write R(x) = p(x) * x. 
We are not given an explicit expression for p(x), but we are given some information about it. In particular, we know that p(40) = 90, which means that the price per widget is 90 dollars when the number of widgets sold is 40. We can use this information to find an expression for p(x). 
Let's assume that p(x) is a linear function of x, so we can write p(x) = mx + b, where m is the slope and b is the y-intercept. We know one point on the line, (40, 90), so we can use that to find m and b. 
First, we can find the slope: 
m = (90 - p(0)) / (40 - 0) = 90/40 = 9/4. 
Next, we can use the point-slope form of a line to write an equation for the line: 
p(x) - 90 = (9/4)(x - 40) 
Simplifying, we get: 
p(x) = (9/4)x - 45. 
Now we can find R(x): 
R(x) = p(x) * x = (9/4)x^2 - 45x. 
Substituting x = 100, we get R(100) = (9/4)(100)^2 - 45(100) = 1575. 
Finally, we can find P(100): 
P(100) = R(100) - C(100) = 1575 - 313 = 1262. 
Therefore, the answer is D) 1687.

Solve the Problem. -In Economics, Functions That Involve Revenue, Cost and Profit Are