Question

Suppose that the price p, in thousand pesos, and the number of sales x(in hundreds) of a certain item can be modeled by the equation 5p+4x+px=100. Suppose also that the price is increasing at the rate of $200 per year. How fast is the quantity changing at the instant when the price of the item is $1500?

Answer 1

The rate of $200 per year is dp/dt,
so we need to differentiate with respect to t

5p+4x+px=100
5 dp/dt + 4 dx/dt + p dx/dt + x dp/dt = 0

we need x when p = 1.5
in the original:
5(1.5) + 4x + 1.5x = 100
5.5x = 92.5
x = 185/11

so we have: p = 1.5, x = 185/11 , dx/dt = .2, and dp/dt = ???

5 dp/dt + 4 dx/dt + p dx/dt + x dp/dt = 0
5dp/dt + 4(.2) + 1.5(.2) + (185/11)(dp/dt) = 0
factor out dp/dt, and finish the arithmetic.
remember that dp/dt will be in thousands of pesos per year.