SOLUTION: HiPlease can somebody help with this problem i) The variables x and y are given in terms of the parameter t as x = 3t(1 + t^3) and y = 3t^2(1 + t^3). Provethat dy/dx = t(2 + 5t^3)/

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons  -> Quadratic Equation Lesson -> SOLUTION: HiPlease can somebody help with this problem i) The variables x and y are given in terms of the parameter t as x = 3t(1 + t^3) and y = 3t^2(1 + t^3). Provethat dy/dx = t(2 + 5t^3)/      Log On


   

Question 375008: HiPlease can somebody help with this problem i) The variables x and y are given in terms of the parameter t as x = 3t(1 + t^3) and y = 3t^2(1 + t^3). Provethat dy/dx = t(2 + 5t^3)/1 + 4t^3 ii) A cycloid is a curve traced out by a point on the rim of a wheel as it rolls along the ground. Its curve maybe given parametrically by x = a(t - sin t) and y = a(1 - cos t), where a is a constant. Show thatdy/dx = cot(t/2) Thanks
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
x+=+3t%281+%2B+t%5E3%29=3t%2B3t%5E4
dx%2Fdt=3%2B12t%5E3
.
.
.
y+=+3t%5E2%281+%2B+t%5E3%29=3t%5E2%2B3t%5E5
dy%2Fdt=6t%2B15t%5E4
.
.
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=%286t%2B15t%5E4%29%2F%283%2B12t%5E3%29
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=%283t%282%2B5t%5E3%29%29%2F%283%281%2B4t%5E3%29%29

.
.
.
Use the same method.
x+=+a%28t+-+sin+t%29
y+=+a%281+-+cos+t%29
.
.
dx%2Fdt=a%281-cos%28t%29%29
dy%2Fdt=a%28sin%28t%29%29
.
.
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=%28a%28sin%28t%29%29%29%2F%28a%281-cos%28t%29%29%29
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=1%2F%281%2Fsin%28t%29-1%2Ftan%28t%29%29
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=1%2F%28csc%28t%29-cot%28t%29%29
dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=1%2Ftan%28t%2F2%29
highlight_green%28dy%2Fdx=%28dy%2Fdt%29%2F%28dx%2Fdt%29=cot%28t%2F2%29%29