Raymond B. answered • 03/28/22
Math, microeconomics or criminal justice
construct a right triangle, hypotenuse is the distance directly to the other side = sqr(80^2 + (250-x)^2) = sqr(6400 + 250^2 -500x + x^2)
the other two sides are 80 m and 250-x meters where x is the distance he runs along the shore line after swimming diagonaly to the opposite side.
time = distance divided by rate of speed
total time = T = x/5 + d/2 = x/5 + sqr(12650 -500x + x^2)/2
T' = 1/5 -(1/4)/sqr(12650-500x+x^2) = 0
sqr(12650 -500x +x^2) =4/5
12650 -500x +x^2 = 16/25
x^2 -500x + 12650-16/25 = 0
x^2 -500x + 12,649.36 = 0
x = 500/2 - (1/2)sqr(250,000 -50,597.44)
= 250 - 223.27 = 26.73 meters along the shore line
250-x = 250-26.73 = 223.27
distance directly across the river squared = d^2 = (223.27)^2 + 80^2 = 237.17^2
d= 237.17 meters
going directly across the river would mean total time = sqr(250^2 + 80^2)/2 = 262.49/2 = 131.24 seconds
going straight across the river and then down 250 meters would mean total time = 80/2 + 250/5 = 40+50= 90 seconds
somewhere in between is the minimum time = less than 90 seconds.
construct a right triangle with sides 80 and 250-x where x is the distance he runs along the shoreline
It helps to draw a picture
then d= the distance he swims across the river with d^2 =square root of ( (250-x)^2 + 80^2)
d = sqr(x^2 -500x + 68900)
Total Time = T = x/5 + d/2 time = distance divided by speed
T = x/5 + (1/2)sqr(x^2-500x +68,900)
take the derivative and set = 0
T' = dT/dx = 1/5 + (1/4)(2x-500)/sqr(x^2-500x + 68,900) = 0
(2x-500)/4(sqr(x^2 -500x +68,900) = -1/5
cross multiply
5x -1250 = -2sqr(x^2 -500x + 68,900)
square both sides
25x^2 -12,500x +1,562,500 = 4x^2 -2,000x + 273,600
21x^2 -10,500x + 1,288,900
use the quadratic formula
x = 216.5
T = x/5 + d/2 = 43.3 + 43.37 = 86.37 seconds
