Question

In planning a restaurant, it is estimated that a profit of $10 per seat will be made if the number of seats is no more than 40 inclusive. On the other hand, the profit on each seat will decrease by 20 cents for each seat above 40. a)Find the number of seats that will produce the maximum profit.

b) What is the maximum profit?

2) A store sells 180,000 cases of a product annually. It costs $12 to store 1 case for 1 year and $10 to produce 1 lot. Find the number of lots that should be ordered each time. Show your work and give your answer as a complete sentence.

3) For a certain good we have q = f(p) = 200e^(-0.4p)
a) Find the elasticity of demand at price p = $50.
b) At p = $50, is the demand elastic, inelastic, or does it have unit elasticity? Explain what this means for this product.
c) Find the elasticity of demand at price p = $20.
d) At p = $20, is the demand elastic, inelastic, or does it have unit elasticity? Explain what this means for this product.

Answer 1

1) a) 45 seats; b) $ 40

2) 548 cases

3) a) 20; b) elastic; c) 8; d) elastic

Explanation:

1) Maximize profit

a) Number of seats to maximize profit

Let p = the profit per seat

and s = the number of seats

We have the conditions:

`p = \begin{cases}10 & \quad 0 \leq s \leq 40\\10 - 0.20(s - 40) & \quad s > 40\\\end{cases}`

The total profit P is the number of seats times the profit per seat.

Thus, the total profit function is

`P = \begin{cases}10s & \quad 0 \leq s \leq 40\\s[10 - 0.20(s - 40)] & \quad s > 40\\\end{cases}`

The equation for the function when s > 40 is

P = s[10 - 0.20(s - 40)] = s(10 - 0.20s + 8) = s(18 - 0.20s) = 18s - 0.20s²

This is the equation of a parabola.

In standard form,

P = -0.20s² + 18s

a = -0.20; b = 18; c = 0

The parabola opens downwards, because a < 0. Therefore, the vertex is a maximum.

`s = - (b)/(2a) = - (18)/(2* (-0.20)) = \mathbf{45}`

The maximum profit occurs at 45 seats.

b) Maximum profit

P = -0.20s² + 18s = -0.20×45² + 18×45 = -405 + 810 = 405

The graph below shows that the maximum profit is $405.

2. Number of lots to order

The number of cases to be ordered to maximize profit is the economic order quantity (q).

The formula is

`q = \sqrt{(2Fm)/(k)`

where

F = the fixed setup cost to manufacture the product

m = the total number of cases produced annually

k = the cost of storing one case for one year

Data:

F = $10/lot

m = 180 000 cases/yr

k = ($12/case)/yr

Calculation:

`\begin{array}{rcl}q &= &\sqrt{(2Fm)/(k)}\\\\& = & \sqrt{(2* 10 * 180000)/(12)}\\\\& = & √(300000)\\& = & \mathbf{548}\\\end{array}\\\text{The company should order lots of $\large \boxed{\textbf{548 cases}}$ each time.}`

3) Elasticity of demand

Elasticity of demand (E) is an indicator of the impact of a price change on a product's sales.

The general formula for an exponential demand curve is

`y = ae^(-bp)`

Your demand curve has the formula

`y = 200e^(-0.4p)`

The formula for elasticity of demand is

`E = -(p)/(q)\frac{\text{d}q}{\text{d}p}`

a) Elasticity at p = $50

The formula for elasticity at p = $50 is

`q(50) = 200e^(-0.4 * 50) = 200e^(-20)\\(p)/(q) = (50)/(200e^(-20))\\\\\frac{\text{d}q}{\text{d}p} = -0.4* 200e^(-0.4p) = -80e^(-20) \\\\E = -(p)/(q)\frac{\text{d}q}{\text{d}p} = -(50)/(200e^(-20)) *\left(-80e^(-20)\right ) = \textbf{20}`

b) Meaning of E for p = $50

E > 1, so the demand is elastic.

If E = 20, a 1 % increase in price causes a 20 % decrease in demand.

c) Elasticity at p = $20

`q(20) = 200e^(-0.4 * 20) = 200e^(-8)\\(p)/(q) = (20)/(200e^(-8))\\\\\frac{\text{d}q}{\text{d}p} = -0.4* 200e^(-0.4p) = -80e^(-8) \\\\E = -(p)/(q)\frac{\text{d}q}{\text{d}p} = -(20)/(200e^(-8)) *\left(-80e^(-8)\right ) = \textbf{8}`

d) Meaning of E for p = $20

E > 1, so the demand is elastic.

If E = 8, a 1 % increase in price causes an 8 % decrease in demand.