Answer:
2.19 km
Step-by-step explanation:
If x is the distance she walks down the road before turning, then the total time is:
t = x/8 + √((3 − x)² + 2²) / 3
t = x/8 + √(9 − 6x + x² + 4) / 3
24t = 3x + 8√(13 − 6x + x²)
24t = 3x + 8(13 − 6x + x²)^½
Take derivative of both sides with respect to x.
24 dt/dx = 3 + 4(13 − 6x + x²)^-½ (-6 + 2x)
When t is a minimum, dt/dx = 0.
0 = 3 + 4(13 − 6x + x²)^-½ (-6 + 2x)
-3 = 4(13 − 6x + x²)^-½ (-6 + 2x)
3 / (6 − 2x) = 4(13 − 6x + x²)^-½
3 / (24 − 8x) = (13 − 6x + x²)^-½
(24 − 8x) / 3 = (13 − 6x + x²)^½
(24 − 8x)² / 9 = 13 − 6x + x²
576 − 384x + 64x² = 117 − 54x + 9x²
459 − 330x + 55x² = 0
Solve with quadratic formula.
x = [ 330 ± √((-330)² − 4(55)(459)) ] / 2(55)
x = (330 ± √7920) / 110
x = 2.19 or 3.81
Since 0 < x < 3, x = 2.19.
