To find the length of the curve given by the parametric equations r(t) = (5t, 3cos(t), 3sin(t)), we will use the formula for the length of a curve.
The length L of a curve defined by parametric equations from t = a to t = b is given by:
L = ∫ab √((dx/dt)² + (dy/dt)² + (dz/dt)²) dt
In our case, the parameters are:
- x(t) = 5t
- y(t) = 3cos(t)
- z(t) = 3sin(t)
We’ll first compute the derivatives with respect to t:
- dx/dt = 5
- dy/dt = -3sin(t)
- dz/dt = 3cos(t)
Now we plug these into our formula:
L = ∫03 √((5)² + (-3sin(t))² + (3cos(t))²) dt
This simplifies to:
= ∫03 √(25 + 9sin²(t) + 9cos²(t)) dt
Since sin²(t) + cos²(t) = 1, we can further simplify:
= ∫03 √(25 + 9) dt
= ∫03 √34 dt
Now we can compute the integral:
= √34 * t |03
= √34 * (3 – 0)
= 3√34
Thus, the length of the curve from t = 0 to t = 3 is 3√34.