How do you find the length of the curve defined by the parametric equations r(t) = 5t, x(t) = 3 cos(t), y(t) = 3 sin(t), and z(t) = 3t?

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To find the length of a curve defined by parametric equations, we can use the arc length formula. For a curve defined in three dimensions by the parameter t, the length L from t = a to t = b is given by the integral:

L = &integral;ab √ (&frac{dX}{dt})2 + (&frac{dY}{dt})2 + (&frac{dZ}{dt})2 dt

In our case, we have the following components:

  • x(t) = 3 cos(t)
  • y(t) = 3 sin(t)
  • z(t) = 3t

Next, we need to compute the derivatives:

  • dx/dt = -3 sin(t)
  • dy/dt = 3 cos(t)
  • dz/dt = 3

Substituting these derivatives into our arc length formula, we get:

L = &integral;ab √ ((-3 sin(t))2 + (3 cos(t))2 + (3)2) dt

Now, simplifying:

L = &integral;ab √ (9 sin2(t) + 9 cos2(t) + 9) dt

Using the identity sin2(t) + cos2(t) = 1, we can simplify this further:

L = &integral;ab √ (9(1) + 9) dt = &integral;ab √ (18) dt = &integral;ab 3√{2} dt

Finally, this integral evaluates to:

L = 3√{2} (b - a)

Thus, to compute the length of your curve, you will need to determine the values of a and b, which represent the bounds of your interval for t. Once you have those, you can easily compute the length.

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