A particle travels along the curve c(t) = (6t^3-6, 7t^2-4). If time t is measured in seconds and if the coordinates are measured in meters, then what is the particle's speed at time t=5?
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To find the particle's speed at time t = 5, we need to find the derivative of the position vector c(t) with respect to time and then calculate its magnitude.
Step 1: Find the derivative of c(t):
To get the derivative, we need to differentiate each component of the position vector c(t) separately.
The x-component, c₁(t), is given by 6t^3 - 6. Taking the derivative of c₁(t) with respect to t, we get:
c'₁(t) = 18t²
The y-component, c₂(t), is given by 7t^2 - 4. Taking the derivative of c₂(t) with respect to t, we get:
c'₂(t) = 14t
Step 2: Calculate the magnitude of the derivative vector:
Now that we have the derivatives, we can calculate the magnitude of the derivative vector using the formula:
|c'(t)| = √(c'₁(t)² + c'₂(t)²)
Substituting the calculated derivatives into the formula, we have:
|c'(t)| = √((18t²)² + (14t)²)
= √(324t⁴ + 196t²)
Step 3: Evaluate the magnitude at t = 5:
We need to find the magnitude of the derivative vector |c'(t)| at t = 5. Substituting t = 5 into the equation:
|c'(5)| = √(324(5)⁴ + 196(5)²)
= √(324(625) + 196(25))
= √(202500 + 4900)
= √207400
≈ 455.05
Therefore, the particle's speed at time t = 5 is approximately 455.05 meters per second.