math 1325 marginal revenue

Question

The demand equation for handcrafted violins by a certain violin maker can be approximated by

p=22-x 

where p is the price in thousands of dollars and x is the quantity of violins demanded. Find and interpret the marginal revenue for each of the given production levels.

​(a)

x=5

​(b)

x=15

​(c)

x=13

Answer 1

First we need to find what the revenue function is: given that the price function is

p(x)=22-x and x is the quantity of violins demanded

Then the revenue from people buying the violins will be

R(x)=xp(x)=22x-x^2.

Now, the marginal revenue is given by R'(x)

R'(x)=22-2x

a)

R'(5)=22-2(5)=12

So, when 5 violins are demanded, the companies revenue is increasing by $12,000 per violin demanded.

b)

R'(15)=22-2(15)=-8

So, when 15 violins are demanded, the companies revenue is decreasing by $8,000 per violin demanded.

c)

R'(13)=22-2(13)=-4

So, when 13 violins are demanded, the companies revenue is decreasing by $4,000 per violin demanded.