The amount of economic aid E (in millions of dollars) a country receives from the USA is related to the military aid x(in millions of dollars) it is also receiving

Question

The amount of economic aid E (in millions of dollars) a country receives from the USA is related to the military aid x(in millions of dollars) it is also receiving by the function E(x)= 10 + sqrt(5x³ + x²).

The military aid is currently $18 million a year, but is decreasing by $4 million per year; how is economic aid changing?

Answer 1

E(x) = 10 + (5x³+x²)^1/2

Given: dx/dt = -4 and x = 18. Find dE/dt.

dE/dt = (1/2)(5x³+x²)^-1/2(15x²+2x)(dx/dt)

= (1/2)(29484)^-1/2(4896)(-4) = -9792 / √29484 = -544 / √91 ≈ -57.03

Economic aid is decreasing at the rate of approximately $57.03 million per year when x = $18 million and x is decreasing at the rate of $4 million per year.