Consider the parametric equation #x= 10(cost+tsint)# and #y= 10(sint-tcost)#, What is the length of the curve from #0# to #((3pi)/2)#?

Answer 1

#10/2((3pi)/2)^2#

#{(x = 10 (Cos t + t Sint)),(y=10 (Sint - t Cost)):}#
#(ds)/dt = sqrt(((dx)/dt)^2+((dy)/dt)^2#
#dx/dt=10tcost# #dy/dt=10t sint#
#(ds)/dt=10t#
#s=int_(t=0)^((3pi)/2)10tdt = 10/2((3pi)/2)^2#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

To find the length of the curve from ( t = 0 ) to ( t = \frac{3\pi}{2} ), you can use the arc length formula for parametric equations:

[ L = \int_{a}^{b} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} dt ]

Given ( x = 10(\cos t + t\sin t) ) and ( y = 10(\sin t - t\cos t) ), you need to compute ( \frac{dx}{dt} ) and ( \frac{dy}{dt} ), and then integrate from ( t = 0 ) to ( t = \frac{3\pi}{2} ).

[ \frac{dx}{dt} = -10(t\cos t) ] [ \frac{dy}{dt} = 10(t\sin t) ]

Now, plug these into the arc length formula:

[ L = \int_{0}^{\frac{3\pi}{2}} \sqrt{\left(-10(t\cos t)\right)^2 + \left(10(t\sin t)\right)^2} dt ]

[ L = \int_{0}^{\frac{3\pi}{2}} \sqrt{100t^2(\cos^2 t + \sin^2 t)} dt ]

[ L = \int_{0}^{\frac{3\pi}{2}} \sqrt{100t^2} dt ]

[ L = \int_{0}^{\frac{3\pi}{2}} 10t dt ]

[ L = 10\int_{0}^{\frac{3\pi}{2}} t dt ]

[ L = 10\left[\frac{t^2}{2}\right]_{0}^{\frac{3\pi}{2}} ]

[ L = 10\left(\frac{9\pi^2}{8}\right) ]

[ L = \frac{45\pi^2}{4} ]

So, the length of the curve from ( t = 0 ) to ( t = \frac{3\pi}{2} ) is ( \frac{45\pi^2}{4} ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7