Test review please help

Question

Answer 1

(12)

The Demand function is p = 60/√x

The cost function is C = 0.65x + 400

The Profit Function is given as Profit P(x) = (Number of Units sold)*(price of the product) - (Cost of the product)

P(x) = x * (60/√x) - (0.65x + 400)

P(x) = 60√x - 0.65x - 400

Apply Derivative on both sides.

P'(x) = 60/(2√x) - 0.65

P'(x) = 30/(√x) - 0.65

Now Marginal Profit P' at x = 100

P'(100) = 30/(√100) - 0.65

P'(100) = 30/10 - 0.65

P'(100) = 3 - 0.65

P'(100) = 2.35

Therefore marginal profit at x = 100 is 2.35 units of currency .

Answer 2

(13)

The fare charge is P = $ (3 - x/40)².

The Bus can hold 60 people maximum.

Let x be the number of people in the bus.

Revenue = (Number of people in the bus)*(fare charged per people)

Let x be the number of people in the bus

Revenue = x * (3 - x/40)²

R = x * (3 - x/40)²

Apply Derivative on both sides.

R' = x(2(3-x/40)*(-1/40) + (3 - x/40)² * 1

R' = -(x/20)(3-x/40) + (3 - x/40)²

Marginal Revenue R' = 0

-(x/20)(3-x/40) + (3 - x/40)² = 0

(3 - x/40)(-x/20 + 3 - x/40) = 0

(3 - x/40) (3 - 3x/40) = 0

(3 - x/40) = 0 or (3 - 3x/40) = 0

x = 120 or x = 40

We know that bus can hold max 60 peoples then x  = 40.

Therefore number of people in the bus is 40.