Please use the correct numbers that are given! Acme Pizza sells one and only one
ID: 3197461 • Letter: P
Question
Please use the correct numbers that are given!
Acme Pizza sells one and only one item for dessert, rice pudding. It is quite popular, but is definitely a side-line as far as Acme is concerned. Nevertheless, Acme is a business, and would like to make the most out of its rice pudding sales A few months ago Acme raised the price of the rice pudding from $1.19 to $1.29. As would be expected, sales fell a little, from an average of 535 per day to an average of 487 per day. After "doing the math the manager believes that the profit from selling the rice pudding actually increased, even though the number sold went down. He wants you to check his math, and recommend to him whether he should increase the price even further. The profit from selling the rice pudding is calculated by simply multiplying the number sold by the price per unit (this is the revenue) and then subtracting the cost of the rice pudding to the store (the cost). The cost of the rice pudding is 49 cents per unit. See the formulas at the top of the next page I. Analvsis 1. Check the manager's math. Does the profit on the rice pudding increase even though the number of units sold went down? 2. The manager feels that if he raises the price by another 10 cents, the profit should increase by the same amount. Explain with a graph what this last statement means and why this amounts to expecting there to be a linear relationship between price and profit, and discuss whether or not you think it is reasonable to make this assumption. By thinking about what would happen if the store raises the price too high (no sales) or decreases it too much (what happens if the price drops to equal the unit cost?), make an argument that the actual function giving profit as a function of price should be concave down, and should therefore have a maximum 3. In order to find that maximum profit, proceed as follows. Instead of a linear relationship between profit and price, assume that there is a linear relationship between demand (the number of units sold per day, on average) and the price. Use the given data to find the linear model. Use this in turn to come up with a model for the profit as a function of price. Use calculus to find the price that gives the maximum profit, if your model turns out to have one. Don't forget the usual check that you have indeed found a maximum.Explanation / Answer
Ans 1.
Initial Profit = 1.19*535 - 0.49*535 = $374.5 per day
Profit [after the price increased] = 1.29*487 - 0.49*487 = $389.6 per day
Yes, the maths was correct, the profit increased despite the reduction in sale
Ans 2.
If the manager assumes that further increment by 10 cents would increase the profit by the same amount, then he is assuming a linear relation between the price and the profit, as in the first case increment by 10 cents resulted in profit. It is certainly not true, let us see the extreme cases, if the price is too high, then no one would purchase the pudding, thus no profit. On the contrary if the price is too low, then the profit would be non. So the rekation has to concave down, so as to have a maxima of profit at the reasonable price and the profit at both, very high and very low price is almost zero.
Ans 3.
Let the price be x
=> demand = -480(x-1.29) + 487 = -480x + 1106.2 [assuming linear relationship and the given data]
=> Profit = [price - unit cost] * demand
=> P(x) = (x-0.49)[-480x + 1106.2]
for maxima, dP(x)/dx = 0
Now, dP(x)/dx = -960x+1341.4 = 0
=>. x = $1.397
Since, double diff of proft function is negative, it is point of maxima