Question

PENCIL SQUARE PRISM A pencil square prism (made from the MacGu?n Tree) with a capacity of 96in3 is to be constructed in the shape of a rectangular box with a square base and an open top. The material for all the sides costs 1.37$/in2 and the material for the base costs 4.11$/in2. The marketing department has determined that the demand for these pencil square prisms is represented by the following function, where p denotes the cup's unit price (in dollars) and x denotes the quantity demanded. p=701.44−43.84x; (0≤x≤15) To get the most out of the MacGu?n Tree we want to use the optimal dimensions of the square prism to reduces the cost for production. No matter how many square prisms, we expect to spend the same amount as the minimum cost to make (this means the cost function will be linear). The boss want to know a few thing about the production of at most 15 of the MacGu?n Tree pencil square prisms.

1 (1) Ignoring the completeness of the square prism,

what is the interval for which the : (a) cost is increasing?

(b) revenue is increasing?

(c) profs?t is increasing?

(d) cost is decreasing?

(e) revenue is decreasing?

(f) profit?t is decreasing?

(2) Ignoring the completeness of the square prism,

what is the: (a) number of units to achieve maximum revenue?

(b) most revenue that can be achieved?

(c) number of units to achieve maximum profi?t?

(d) most pro?t that can be achieved?

(3) In this situation does the most revenue imply the most profi?t?

(4) After how many units need to be sold before we start to loss a pro?fitt?

Answer 1

Given, Cost of the material of all sides of sq. prism = $ 1.37 in²

  Cost of the material of the base = $ 4.11 in²

Total Cost of the material of a mac tree pencil = all sides + base + top = 1.37 + 4.11 + 4.11 = $ 9.59

Given , Cup price unit, P = 701.44 - 43.48 X, ( X = Quantity demand ,( 0 $\leq$ X $\leq$ 15)

Now, Let see   Cost of material mac tree pencil Cost of Cup unit,(P)

when X = 1 9.59 657.64

2 19.18 613.76

3 28.77 569.92

12 115.08 175.36

13 124.67 131.52

14 134.26 87.68

15 143.85 43.84

From above we can find

1. a) When cost is increasing, P decreases at X = (1 $\leq$ 15).

b) When revenue is increasing at X = (14,15)

c) When profit is increasing at X = (14,15)

d) Cost is decreasing when X decreases and P  is increases

e) Revenue is decreasing when X decreases and P increases

f) Profit is decreasing when X decreases and P increases

2. a) To achieve maximum revenue when X = 15

b) Most revenue can be achieved when X = (14,15)

c) To achieve maximum profit when X = 15

d) Most profit can be achieved when X = 15

3. Yes, Most revenues imply most profit at X=15

4. Minimum unit to be sold before start to losing at X = 14,15