To find the values of t for which the curvature is largest, we need to find the maximum of the curvature function K(t).
The curvature formula is K(t) = ||r'(t) x r''(t)|| / ||r'(t)||^3, where r'(t) is the first derivative of r(t), r''(t) is the second derivative of r(t), and ||v|| represents the magnitude of the vector v.
Let's go through the calculations step by step to see where the discrepancy lies.
First, find r'(t):
r'(t) = <1 - 3cos(t)/2, 3sin(t)/2, 1>
Next, find r''(t):
r''(t) = <3sin(t)/2, 3cos(t)/2, 0>
Now, calculate r'(t) x r''(t):
r'(t) x r''(t) = (-3cos(t)/2)(3cos(t)/2) - (3sin(t)/2)(3sin(t)/2) = -9/4*cos^2(t) - 9/4*sin^2(t) = -9/4
The magnitude of r'(t) x r''(t) is ||r'(t) x r''(t)|| = |-9/4| = 9/4.
Next, calculate ||r'(t)||:
||r'(t)|| = sqrt((1 - 3cos(t)/2)^2 + (3sin(t)/2)^2 + 1^2) = sqrt(1 - 3cos(t) + 9cos^2(t)/4 + 9sin^2(t)/4 + 1) = sqrt(9cos^2(t)/4 - 6cos(t) + 17/4)
Finally, calculate K(t):
K(t) = (9/4) / (sqrt(9cos^2(t)/4 - 6cos(t) + 17/4))^3 = (9/4) / (9cos^2(t)/4 - 6cos(t) + 17/4)^(3/2)
Now, let's compare this result with the given answer key:
The given answer key is [√(4cos^2(t) - 12cos(t) + 13)] / [17 - 12cos(t)]^(3/2).
To simplify this expression, notice that √(4cos^2(t) - 12cos(t) + 13) = √[(2cos(t) - 3)^2 + 4] = sqrt(17 - 12cos(t)). Hence, the given answer key can be rewritten as sqrt(17 - 12cos(t)) / [17 - 12cos(t)]^(3/2).
Comparing this with our result, it seems that there is a discrepancy related to the expression inside the square root.
Please recheck your calculations for the vector cross product and the magnitude of r'(t) x r''(t), as well as any simplifications made along the way. Make sure that the vector calculations are executed correctly and that the expressions are simplified accurately.