Answer:
- T(x) = (√(4+x^2))/2 +(c -x)/4
- (a) x = 1.155 miles
- (b) x = 1/2 mile
Step-by-step explanation:
The straight line distance from the starting point to the point x on the road is given by the Pythagorean theorem:
d1 = √(2²+x²)
The time required to walk that distance in the woods (at 2 miles per hour) is ...
time = distance / speed
t1 = √(4+x²)/2
The remaining distance along the road to the car is ...
d2 = c - x . . . . . for c > x
and the corresponding time at 4 mph is ...
t2 = (c -x)/4
Then the total time is ...
T(x) = t1 +t2
T(x) = √(4+x²)/2 +(c -x)/4
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(a) The value of x that minimizes total time is the one that makes the derivative zero.
T'(x) = x/(2√(4+x²)) -1/4
0 = (2x -√(4+x²))/(4√(4+x²))
√(4+x²) = 2x . . . . . fraction is zero when numerator is zero; add radical
4 +x² = 4x² . . . . . . square both sides
4/3 = x² . . . . . . . . . subtract x², divide by 3
x = (2/3)√3 ≈ 1.1547
If c = 9 miles, x = 1.1547 miles.
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(b) For c < (2/3)√3, the shortest travel time will be along the straight-line path to the car.
If c = 1/2 mile, x = 1/2 mile.
