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.
- Steve4PhysicsLv 72 weeks 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