Final answer:
To find the length of the path of c(t) = (1 + 2t, 2 + 4t) over the interval 1 ≤ t ≤ 4 using the arc length formula, we integrate the magnitude of its derivative.
Step-by-step explanation:
To find the length of the path of c(t) = (1 + 2t, 2 + 4t) over the interval 1 ≤ t ≤ 4 using the arc length formula, we need to integrate the magnitude of the derivative of c(t) with respect to t over the given interval.
The derivative of c(t) is c'(t) = (2, 4), and its magnitude is √(2^2 + 4^2) = √20 = 2√5.
Now, we integrate 2√5 with respect to t from 1 to 4:
∫14 2√5 dt = 2√5t2/2∣14 = √5(42 - 12)/2 = √5(16 - 1)/2 = √5(15)/2 = (15√5)/2
Therefore, the length of the path is (15√5)/2 units.