Anonymous asked in Science & MathematicsMathematics · 8 months ago

Multivariable Calculus Help(Use Line Integral)?

Farmer Pickles wants Bob to paint the circular fence which encloses his sunflower field. If the parametric equations x= 18 cos(θ) and y= 18 sin(θ) describe the base of the fence (in yards) and the height of the fence is given by the equation h(x,y) = 12 + (2x−y)/6, then how many gallons of paint will Bob need to complete the project. Assume that one gallon of paint covers three hundred square feet of fence.

1 Answer

  • 8 months ago

    This is not a line integral problem – it can but done using calculus as a surface area integral problem.  However, there is a simple non-calculus solution.

    x= 18 cos(θ) and y= 18 sin(θ) gives a circle radius 18 yards. Circumference c = 2πr = 36π yards.

    h =  12 + (2x−y)/6 (presumably also in yards – but that’s a very high fence!)

    = 12 + (2(18cos(θ)) − 18sin(θ))/6

    Save yourself a lot of work by noting that the average of cos(θ) and sin(θ) over 0≤ θ<2π is zero.  (Think of the graphs from sin and cos – half positive and half negative.)

    That means average value of h is: h_avg = 12yards

    Area: A = C*h_avg = 36π * 12 = 432π sq. yds.

    =  432π * 9 sq ft = 3888π  sq ft

    Amount of paint = 3888π /300 = 40.7 gallons

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