Calculate the consumer surplus (in dollars) at the unit price \( \bar{p}=90 \) for the demand equation \( p=670 e^{-4 q} \). (Round your answer to the nearest cent.)

We are given the demand equation   p = 670 · e^(–4q) and the market price (unit price)   p̅ = 90. Step 1. Find the quantity demanded, q*, at p̅ = 90. Set the price equal to 90:   90 = 670 · e^(–4q*). Solve for q*:   e^(–4q*) = 90/670  ⟹  –4q* = ln(90/670),   q* = –(1/4) · ln(90/670). It is often easier to write this as:   q* = (1/4) · ln(670/90). Step 2. Write the consumer surplus formula. Consumer surplus (CS) is the area under the demand curve from 0 to q* minus the total amount paid at the market price:   CS = ∫₀^(q*) [demand function] dq – p̅ · q*     = ∫₀^(q*) 670 · e^(–4q) dq – 90 · q*. Step 3. Evaluate the integral. Compute the indefinite integral of 670 · e^(–4q):   ∫ 670 · e^(–4q) dq = 670 · (–1/4) e^(–4q) + C = –(670/4) e^(–4q) + C. Now, evaluate from 0 to q*:   ∫₀^(q*) 670 · e^(–4q) dq = [–(670/4) e^(–4q)]₀^(q*)     = [–(670/4) e^(–4q*)] – [–(670/4) e^(0)]     = (670/4)[1 – e^(–4q*)]. But notice that from the equation for q*, we have:   e^(–4q*) = 90/670. Thus, the integral becomes:   ∫₀^(q*) 670 · e^(–4q) dq = (670/4)[1 – (90/670)]     = (670/4) · (580/670)     = 580/4 = 145. Step 4. Compute q* numerically. Remember, q* = (1/4) ln(670/90). First, compute the ratio:   670/90 ≈ 7.4444. Now, take the natural logarithm:   ln(7.4444) ≈ 2.007. Thus,   q* ≈ (1/4) · 2.007 ≈ 0.50175. Step 5. Calculate p̅ · q*.   90 · q* ≈ 90 · 0.50175 ≈ 45.1575. Step 6. Find the consumer surplus.   CS = (Area under demand curve) – (Total expenditure)     ≈ 145 – 45.1575     ≈ 99.8425. Rounding to the nearest cent, the consumer surplus is approximately $99.84. Final Answer: The consumer surplus at a unit price of $90 is about $99.84.