The demand equation for the Sicard sports watch is given by where x (measured in units of a thousand) is the quantity demanded per week and p is the unit price in dollars. Use the Trapezoidal Rule (take n = 8) to estimate the consumers' surplus if the market price is $25/watch. Round answer to two decimal places.
A) $392,373.56
B) $142.37
C) $142,373.56
D) $367,373.56

Question

The demand equation for the Sicard sports watch is given by  where x (measured in units of a thousand) is the quantity demanded per week and p is the unit price in dollars. Use the Trapezoidal Rule (take n = 8) to estimate the consumers' surplus if the market price is $25/watch. Round answer to two decimal places.
A) $392,373.56 
B) $142.37 
C) $142,373.56 
D) $367,373.56

Quizplus · Accepted Answer

To use the Trapezoidal Rule, we first need to find the inverse demand function (expressing p in terms of x):
p = 11e^(8/x)
Setting p = 25 and solving for x, we get x ≈ 4.3866 (measured in thousands).

Next, we need to set up the integral for consumer surplus:
CS = ∫_0^4.3866 (11e^(8/x) - 25) dx

We will use n = 8 subintervals of equal width:
Δx = (4.3866 - 0)/8 = 0.5483 (measured in thousands)

Using the Trapezoidal Rule formula, we get:
CS ≈ Δx/2 [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_7) + f(x_8)]
where f(x) = 11e^(8/x) - 25 and x_i = iΔx for i = 0, 1, 2, ..., 8.

Calculating the values:
f(x_0) = f(0) = 11e^∞ - 25 = -25
f(x_1) = f(0.5483) ≈ 5.05
f(x_2) = f(1.0966) ≈ 10.97
f(x_3) = f(1.6449) ≈ 17.99
f(x_4) = f(2.1932) ≈ 25.98
f(x_5) = f(2.7415) ≈ 34.61
f(x_6) = f(3.2898) ≈ 44.28
f(x_7) = f(3.8381) ≈ 55.41
f(x_8) = f(4.3866) ≈ 68.30

Plugging in these values, we get:
CS ≈ 0.5483/2 [-25 + 2(5.05) + 2(10.97) + 2(17.99) + 2(25.98) + 2(34.61) + 2(44.28) + 2(55.41) + 68.30]
CS ≈ $142,373.56 (rounded to two decimal places)

Therefore, the best choice is C.

The Demand Equation for the Sicard Sports Watch Is Given