Math Snap

STEP 1

1. We are given the supply equation $p = 110 + e^{0.01q}$.
2. The unit price is given as $\bar{p} = 150$.
3. We need to calculate the producers' surplus at this unit price.
4. The producers' surplus is the area above the supply curve and below the price line from $q = 0$ to the equilibrium quantity.

STEP 2

1. Determine the equilibrium quantity $q$ at the given unit price $\bar{p}$.
2. Set up the integral for the supply curve from $q = 0$ to the equilibrium quantity.
3. Calculate the integral to find the total revenue.
4. Calculate the producers' surplus using the formula: Producers' Surplus = Total Revenue - Integral of the supply curve.

STEP 3

Set the supply equation equal to the unit price to find the equilibrium quantity:
$$150 = 110 + e^{0.01q}$$

STEP 4

Solve for $q$:
$$150 - 110 = e^{0.01q}$$ $$40 = e^{0.01q}$$ Take the natural logarithm of both sides:
$$\ln(40) = 0.01q$$ Solve for $q$:
$$q = \frac{\ln(40)}{0.01}$$ Calculate $q$:
$$q \approx \frac{3.6889}{0.01}$$ $$q \approx 368.89$$

STEP 5

Set up the integral for the supply curve from $q = 0$ to $q = 368.89$:
$$\int_{0}^{368.89} (110 + e^{0.01q}) \, dq$$

STEP 6

Calculate the integral:
$$\int_{0}^{368.89} (110 + e^{0.01q}) \, dq = \left[ 110q + \frac{1}{0.01}e^{0.01q} \right]_{0}^{368.89}$$ Evaluate the integral:
$$= \left[ 110(368.89) + 100e^{0.01 \times 368.89} \right] - \left[ 110(0) + 100e^{0.01 \times 0} \right]$$ $$= \left[ 40577.9 + 100 \times 40 \right] - \left[ 0 + 100 \right]$$ $$= 40577.9 + 4000 - 100$$ $$= 44477.9$$

Calculate the producers' surplus:
Producers' Surplus = Total Revenue - Integral of the supply curve
Total Revenue = $\bar{p} \times q = 150 \times 368.89$
$$= 55333.5$$ Producers' Surplus:
$$= 55333.5 - 44477.9$$ $$= 10855.6$$ Round to the nearest cent:
$$\boxed{10855.60}$$