Find the roots of the equation x^2 - x - 6
the roots of the equation are the values of x that make the equation equal to zero. so we want to solve x^2 - x - 6 = 0 We can factor it into the form (x + some number)*(x = some number) Then we know the numbers: 1) have to equal -6 when multiplied together 2) have to add up to -1 so to equal -6, we have the options 1 and -6, -1 and 6, 2 and -3, 3 and -2 looking at these pairs of numbers and adding them we see that only 2 and -3 fulfill the requirement that they add up to -1 So we can factor this equation as follows, (x + 2)*(x - 3) and this equals zero if x +2 = 0 or x - 3 = 0 so x + 2 = 0 if x = -2 and x - 3 = 0 if x = 3 So the two roots of this equation are -2 and 3
You want to build a rectangular enclosure out of fencing that maximizes the area enclosed by the four sides subject to the constraint that you have 40 feet of fencing to construct the enclosure.
for a rectangle, the area is X*Y the constraint is 2X + 2Y = 40 So the problem is maximize X*Y subject to 2X + 2Y = 40 We want to rearrange to constraint and get rid of it by substituting it into what we are maximizing 2X + 2Y = 40 X + Y = 20 by dividing through by 2 X = 20 - Y substitute this into X*Y (20 - Y)*Y 20Y - Y^2 so we now have this in terms of one variable and can optimize it by taking first order conditions( meaning we take the first derivative of it and set it equal to zero). d/dY of 20Y - Y^2 is 20 - 2Y set it equal to zer0, 20 - 2Y = 0 and solve: 20 = 2Y, or Y = 10 (and technically we should confirm this is a mazimum by way of the second derivative test, the second derivate of 20Y - Y^2 is -2, and since it is negative this is a maximum) Then we can plug in Y=10 to the original constraint to find X 2X + 2*(10) = 40 and solving this gives us X = 10 So the answer is that a square with sides of length 10 feet optimizes the area enclosed.
How do externalities cause inefficiencies in a market? Provide an example.
The efficient level of production occurs at the quantity where marginal cost is equal to marginal benefit. When a negative externality exists, this causes the social marginal cost to be greater than the private marginal cost experienced by the firm or individual. This causes the level of production, or consumption, to be greater than what is efficient. When there is a positive externality, this causes the social marginal benefit to be greater than the private marginal benefit, causing the level of production, or consumption, to be less than what is optimal or efficient. The classic example of a negative externality is the pollution caused a firm producing some good. Their private marginal costs includes the cost associate with producing an extra unit along with any detriment they experience due to the pollution. But other individuals will be negatively affected by that pollution as well, and that is not included in the private marginal cost. This causes the firm to produce more than they would if they were accountable for that additional cost the pollution inflicts on the public.