Question

the fuel tanks for airplanes are in the wings, cross section below. the tank must hold 6000 lb of fuel with density 42 lb/ft³. estimate the length of the tank using simpsons rule
horizontal spacing = 1.8 ft
y₀ = 1.3 ft, y₁ = 1.5 ft, y₂ = 1.8 ft, y₃ = 2 ft, y₄ = 2.1 ft, y₅ = 2.3 ft, y₆ = 2.1 ft
the length of the tank is □ ft.
(round to the nearest tenth as needed.)

Explanation:

Step1: Calculate the area using Simpson's Rule

Simpson's Rule formula for \(n = 6\) (even) is \(A=\frac{\Delta x}{3}(y_0 + 4y_1+2y_2 + 4y_3+2y_4+4y_5 + y_6)\), where \(\Delta x = 1.8\) ft.
Substitute \(y_0 = 1.3\), \(y_1 = 1.5\), \(y_2 = 1.8\), \(y_3 = 2\), \(y_4 = 2.1\), \(y_5 = 2.3\), \(y_6 = 2.1\) into the formula:

$$
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$$

Step2: Find the volume and then the length

The weight of the fuel \(W=\text{density}\times\text{Volume}\), so \(\text{Volume}=\frac{W}{\text{density}}\). Given \(W = 6000\) lb and \(\text{density}=42\) lb/ft³, \(\text{Volume}=\frac{6000}{42}\approx142.86\) ft³.
Also, \(\text{Volume}=A\times L\) (where \(L\) is the length of the tank). Then \(L=\frac{\text{Volume}}{A}\).
Substitute \(A = 20.64\) and \(\text{Volume}\approx142.86\) into the formula: \(L=\frac{142.86}{20.64}\approx6.9\)

Answer:

\(6.9\)