word problems

Question

demand for an overseas tour is q = 250 reservations per month when the tour is priced at p = $2000, and q = 50 reservations per month when it is priced at p = $4000.

a) Find a linear model for demand q = D(p).

b) Find the percentage change in demand when price is raised form $2000 to $2100 (i.e., price rises 5%). Give the value of the ratio E = (% decrease demand)/(% increase price).

c) Find the total revenue as a function of price, R (p).

d) If the price is currently $2000 per tour, will revenue rise or fall if the price is increased to $2100?

Answer 1

a) The slope is:

m = (y₂ - y₁) / (x₂ - x₁) = (q₂ - q₁) / (p₂ - p₁) = (250 - 50) / (2000 - 4000) = 200 / -2000 = -1/10 .

Using point-slope form, q - q₁ = m (p - p₁) becomes :

q - 50 = -1/10 (p - 4000) = -1/10 p + 400, and therefore:

q = -1/10 * p +450 = D(p)

b) D(2100) = -1/10 * 2100 + 450 = 240

(240 - 250) / 250 = -10/250 = -0.04 = -4% (percentage change in demand for a 5% increase in price)

c) R(p) = p * D(p) = p * (-1/10 * p + 450) = -1/10 * p² +450 * p = R(p) (Revenue as a function of price)

d) R(2000) = -1/10 * (2000)² + 450 * 2000 = 500000

R(2100) = -1/10 * (2100)² + 450 * 2100 = 504000

Hence, revenue will rise if the price is increased from $2000 to $2100.