Question

Find r(t) if r'(t) = 9t^8i + 9t^8j + tk and r(1) = i + j.

A) r(t) = 3t^9i + 3t^9j + 0.5t^2k
B) r(t) = t^9i + t^9j + 0.5t^2k
C) r(t) = 4.5t^9i + 4.5t^9j + 0.5t^2k
D) r(t) = 2t^9i + 2t^9j + 0.5t^2k

Answer 1

After integrating each component of r'(t), applying the initial condition r(1) = i + j, and finding the constants of integration, the correct position vector is r(t) = t^9i + t^9j + 0.5t^2k, which is option B.

So option (B) is the correct answer

Step-by-step explanation:

We are given the derivative of the position vector r'(t) and the initial condition r(1) = i + j. To find the position vector r(t), we need to integrate the derivative with respect to time, t.

Given:

r'(t) = 9t^8i + 9t^8j + tk

Integrate each component with respect to t:

For i component: ∫ 9t^8 dt = t^9 + Ci

For j component: ∫ 9t^8 dt = t^9 + Cj

For k component: ∫ t dt = 0.5t^2 + Ck

Now, apply the initial condition r(1) = i + j to find the constants Ci, Cj, and Ck:

r(1) = (1^9 + Ci)i + (1^9 + Cj)j + (0.5*1^2 + Ck)k = i + j

So Ci = Cj = 0 and Ck = 0

Using the calculated constants, we get:

r(t) = t^9i + t^9j + 0.5t^2k

Therefore, the answer is B) r(t) = t^9i + t^9j + 0.5t^2k.