High School

Effect of Price on Supply of Eggs. Suppose the wholesale price of a certain brand of medium-sized eggs rho (in dollars/carton) is related to the weekly supply x (in thousands of cartons) by the following equation.

625rho² - x² = 100

If 29000 cartons of eggs are available at the beginning of a certain week and the price is falling at the rate of 3$/carton/week, at what rate is the supply changing?

Answer :

Final answer:

The supply of eggs is changing at a rate of approximately 0.0388 cartons/week.

Explanation:

To find the rate at which the supply is changing, we need to differentiate the equation 625rho² - x² = 100 with respect to time (t) and solve for dx/dt.

Given:

  • Initial supply (x) = 29000 cartons
  • Rate of change of price (d(rho)/dt) = -3$/carton/week

Let's differentiate the equation with respect to time:

625(2rho)(d(rho)/dt) - 2x(dx/dt) = 0

Simplifying the equation:

1250rho(d(rho)/dt) - 2x(dx/dt) = 0

Substituting the given values:

1250rho(-3) - 2(29000)(dx/dt) = 0

Simplifying further:

-3750rho - 58000(dx/dt) = 0

Now, we can solve for dx/dt:

dx/dt = (-3750rho) / 58000

Substituting the given rate of change of price:

dx/dt = (-3750rho) / 58000 = (-3750(-3)) / 58000 = 2250 / 58000 = 0.0388 cartons/week

Therefore, the supply is changing at a rate of approximately 0.0388 cartons/week.

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