3. (6 points) Suppose that people are willing to buy s pounds of chocolate candy

Question

3. (6 points) Suppose that people are willing to buy s pounds of chocolate candy per day 180 at Sp per quarter pound, as given by the price demand equation f(p) = x = 10+- (a) Find the demand when the price is $5 (b) Use differentials to approximate the change in demand when the price changes from $5 to $5.5. Note that in this problem, dz represents what dy usually does, and dp represents what dx usually does (since z is a function of p). (c) Find the instantaneous rate of change of the demand with respect to the price when the price is S8. Interpret the result.

Explanation / Answer

x = 10 + 180/p where p is price and 'x' is demand;
a) when p=5$, x= 10 + 180/5 = 10+36 = 46 pounds;
So when price is 5$, the demand is 46 pounds;

b) we have x= 10+180/p
dx/dp = 0 -180/p2
dx/dp = -180/p2
Thus, at p=5,
dx/dp = -180/52 = -7.2
This means that at p=5, a unit change (increase) in p will bring about change (decrease) of 7.2 pounds in x;
But we are concerned with only half unit change in 'p', since p=5.5-p=5 = 0.5;
Thus, for half a unit change we will witness 7.2/5 = 3.6 pounds of change;

Thus, approximating using differentials, a change in p from 5 to 5.5 $ will reduce the demand by approximately 3.6 pounds;

c) dx/dp = -180/p2
Thus, when price =8$, dx/dp = -180/82 = -180/160 = -1.125

Thus, instantaneous rate of change of the demand with respect to the price when the price is 8$ is = -1.125 pounds;
This means that, when price is 8$, any change in the price will bring about a multipler 1.125 change in demand in the opposite direction; i.e. increase in price leading to decarease in demand and vice versa

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