Question

The demand equation for soccer tournament T-shirts is xy − 5,000 = y where y is the number of T-shirts the Enormous State University soccer team can sell at a price of $x per shirt. Find dy dx x = 6 . dy/dx x = 6 = T-shirts per dollar Interpret the result. When the price is set at $6, the demand is by T-shirts per $1 increase in price.

Answer 1

The given equation is:

x y – 5000 = y

Rewrite this to create an explicit equation:

x y – y = 5000

y ( x – 1) = 5000

y = 5000 / (x – 1)

Derive to get dy / dx:

dy = - 5000 dx / (x – 1)^2

dy / dx = - 5000 / (x – 1)^2

So plugging in x = 6,

dy / dx = - 5000 / (6 – 1)^2

dy / dx = - 200

Answer 2

`(dy)/(dx)|_(x=6)= -200`

Explanation:

Given : Demand equation
`xy − 5,000 = y`

To Find
`(dy)/(dx)|_(x=6)`

Solution :

`xy- y= 5000`

`y(x-1)= 5000`

`y= (5000)/(x-1)`

Differentiating with respect to x

`(dy)/(dx)= (-5000)/((x-1)^2)`

Now substitute x = 6

`(dy)/(dx)|_(x=6)= (-5000)/((6-1)^2)`

`(dy)/(dx)|_(x=6)= (-5000)/(5^2)`

`(dy)/(dx)|_(x=6)= (-5000)/(25)`

`(dy)/(dx)|_(x=6)= -200`

Hence
`(dy)/(dx)|_(x=6)= -200` =T-shirts per dollar