(a) Find the consumer's surplus for the demand function p=25−x−x2, where the prevailing market price p0=19. OR (b) Solve the following initial value differential equation : xdxdy=y−xtan(xy), given y=2π when x=1.
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Step-by-Step Solution
Step 1: Find the quantity demanded at market price
To find the quantity demanded (x0) at the prevailing market price (p0=19), we set the demand function equal to the market price and solve for x. This gives us a quadratic equation.
Step 2: Solve the quadratic equation for x
Rearranging the equation 19=25−x−x2 gives x2+x−6=0. Factoring this quadratic equation yields (x+3)(x−2)=0. Since quantity cannot be negative, we take the positive value, x0=2.
Step 3: Apply the consumer's surplus formula
The consumer's surplus (CS) is calculated by integrating the difference between the demand function p(x) and the market price p0 from 0 to the quantity demanded x0.
Step 4: Substitute values and integrate
Substitute the demand function p(x)=25−x−x2 and the market price p0=19 into the formula. Simplify the integrand to 6−x−x2 and then perform the integration.
Step 5: Evaluate the definite integral
Evaluate the definite integral by plugging in the upper limit x=2 and the lower limit x=0. Subtract the value at the lower limit from the value at the upper limit to find the consumer's surplus.