Math  /  Calculus

QuestionCalculate the producers' surplus for the supply equation at the indicated unit price pˉ\bar{p}. HINT [See Example 2.] (Round your answer to the nearest cent.) p=110+e0.01q;pˉ=150$10842.89×\begin{array}{l} \quad p=110+e^{0.01 q} ; \bar{p}=150 \\ \$ 10842.89 \times \end{array}

Studdy Solution

STEP 1

1. We are given the supply equation p=110+e0.01q p = 110 + e^{0.01q} .
2. The unit price is given as pˉ=150 \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 q = 0 to the equilibrium quantity.

STEP 2

1. Determine the equilibrium quantity q q at the given unit price pˉ \bar{p} .
2. Set up the integral for the supply curve from q=0 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+e0.01q 150 = 110 + e^{0.01q}

STEP 4

Solve for q q :
150110=e0.01q 150 - 110 = e^{0.01q} 40=e0.01q 40 = e^{0.01q}
Take the natural logarithm of both sides:
ln(40)=0.01q \ln(40) = 0.01q
Solve for q q :
q=ln(40)0.01 q = \frac{\ln(40)}{0.01}
Calculate q q :
q3.68890.01 q \approx \frac{3.6889}{0.01} q368.89 q \approx 368.89

STEP 5

Set up the integral for the supply curve from q=0 q = 0 to q=368.89 q = 368.89 :
0368.89(110+e0.01q)dq \int_{0}^{368.89} (110 + e^{0.01q}) \, dq

STEP 6

Calculate the integral:
0368.89(110+e0.01q)dq=[110q+10.01e0.01q]0368.89 \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:
=[110(368.89)+100e0.01×368.89][110(0)+100e0.01×0] = \left[ 110(368.89) + 100e^{0.01 \times 368.89} \right] - \left[ 110(0) + 100e^{0.01 \times 0} \right]
=[40577.9+100×40][0+100] = \left[ 40577.9 + 100 \times 40 \right] - \left[ 0 + 100 \right]
=40577.9+4000100 = 40577.9 + 4000 - 100
=44477.9 = 44477.9

STEP 7

Calculate the producers' surplus:
Producers' Surplus = Total Revenue - Integral of the supply curve
Total Revenue = pˉ×q=150×368.89 \bar{p} \times q = 150 \times 368.89
=55333.5 = 55333.5
Producers' Surplus:
=55333.544477.9 = 55333.5 - 44477.9
=10855.6 = 10855.6
Round to the nearest cent:
10855.60 \boxed{10855.60}

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord