The monthly demand for a certain brand of perfume is given by the demand equatio

Question

The monthly demand for a certain brand of perfume is given by the demand equation rho = 100e^-0.0003x + 100 where rho denotes the retail unit price (in dollars) and x denotes the quantity (in 1-oz bottles) demanded. Find the rate of change of the price per bottle when x = 1000 and when x = 3000. (Round your answers to four decimal places.) x = 1000 [0.0200] dollars/bottle x = 3000 [0.0120] dollars/bottle What is the price per bottle when x = 1000' When x = 3000? (Round your answers to the nearest cent.) x = 1000 $[174.00] x = 3000 $ [140.60]

Explanation / Answer

Solution:

(a)
p = 100e-0.0003x + 100differentiate

dp dx = (-0.0003) * 100e-0.0003x = (-0.03) e-0.0003x

evaluated at: x = 1000;

dp{1000} dx = (-0.03) • e-0.0003*1000

dp{1000} dx = -$0.0222 per bottle

evaluated at: x = 3000

dp{3000} dx = (-0.03) • e-0.0003*3000

dp{3000} dx = -$0.0122 per bottle

(b)

p = 100e-0.0003x + 100

evaluated at: x = 1000

p{1000} = 100e-0.0003*1000 + 100

p{1000} = $174.0818 per bottle

evaluated at: x = 3000

p{3000} = 100e-0.0003*3000 + 100

p{3000} = $140.6570 per bottle

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