The time required for a skier, initially at rest, to ski from point A to point B on an inclined plane (where B is not directly downhill from A is given by: . In the equation above, a is the value of the parameter u at point A, b is the value of u at point B, and k is a constant. The expressions x(u) and y(u) represent parametrically the trajectory taken by the skier. The "straight downhill" direction is assumed to be in the direction of increasing values of y.
Find the time required to complete the trip if the skier chooses the following trajectory: .

A) 5.4802k
B) 5.0956k
C) 91.310k
D) 90.2388k

Question

The time required for a skier, initially at rest, to ski from point A to point B on an inclined plane (where B is not directly downhill from A is given by:  . In the equation above, a is the value of the parameter u at point A, b is the value of u at point B, and k is a constant. The expressions x(u) and y(u) represent parametrically the trajectory taken by the skier. The "straight downhill" direction is assumed to be in the direction of increasing values of y.
Find the time required to complete the trip if the skier chooses the following trajectory:  .

A) 5.4802k 
B) 5.0956k 
C) 91.310k 
D) 90.2388k

Quizplus · Accepted Answer

To find the time required to complete the trip, we need to evaluate the integral of the time function given in the problem:
t = ∫[sqrt(1 + (dy/dx)^2)]/[kcos(arctan(dy/dx))] dx from A to B

Using the given parametric equations, we can find that dy/dx = (y(u) - 0)/(x(u) - 0) = y(u)/x(u) and dx/dt = x'(u)/t'(u) = cos(arctan(y(u)/x(u))) / [k(1 + (y(u)/x(u))^2)^(1/2)]

Substituting these expressions in the time integral and simplifying, we get:
t = ∫[sqrt(1 + (y(u)/x(u))^2) * (1 + (y(u)/x(u))^2)^(1/2)] / [k * y(u)] du from a to b

Now, we can substitute the given values of a and b in the above expression and evaluate the integral using a calculator. The final answer is approximately 5.0956k. Therefore, the correct choice is B.

The Time Required for a Skier, Initially at Rest, to Ski