SOLUTION: I've been working on this problem and can only come up with an estimation, with no way of showing how I got there. A person is in a rowboat 3 miles from the closest point on a str

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Question 725594: I've been working on this problem and can only come up with an estimation, with no way of showing how I got there.
A person is in a rowboat 3 miles from the closest point on a straight shoreline, as illustrated in the figure. The person would like to reach a cabin that is 8 miles down the shoreline. The person can row at 4 miles per hour and jog at 7 miles per hour.
A.)How long will it take to reach the cabin if the person rows straight toward shore at point A and then jogs to the cabin?
I estimated 1 hr. 50 minutes, but that is just a guess.
B.) How long will it take to reach the cabin if the person rows straight to the cabin and does no jogging?
C.)Find the minimum time to reach the cabin.
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
A person is in a rowboat 3 miles from the closest point on a straight shoreline, as illustrated in the figure.
The person would like to reach a cabin that is 8 miles down the shoreline. The person can row at 4 miles per hour and jog at 7 miles per hour.
:
A.)How long will it take to reach the cabin if the person rows straight toward shore at point A and then jogs to the cabin?
A time equation: time = dist/speed
total time (t)= row time + walk time
t = 3%2F4 + 8%2F7
t = .75 + 1.143
t 1.893 hrs or about 1 hr + .893(60) = 53.6 min
:
B.) How long will it take to reach the cabin if the person rows straight to the cabin and does no jogging?
Solve this as a right triangle, direct to the cabin will be the hypotenuse
h = sqrt%283%5E2%2B8%5E2%29
h = 8.544 miles
Rowing at 4 mph: 8.544%2F4 = 2.136 hrs or about 2 hrs 8 min
:
C.)Find the minimum time to reach the cabin.
This is a little more complicated. He will row to a point on the shore line, then jog from that point to the cabin.
The position of the landing point has to be determined to find the rowing distance and the jogging distance that will yield the least total travel time>
:
Let x = distance from the closest point on the shoreline to the landing point
Then
(8-x) = distance from the landing point to the cabin (the jogging distance)
The rowing distance will be the hypotenuse of the triangle formed by x and 3
:
total time = rowing time + jogging time
t(x) = %28sqrt%28x%5E2%2B3%5E2%29%29%2F4 + %288-x%29%2F7
:
Graphically

Minimum on the graph, x=2.1 mi from the closest point on shoreline, then
Minimum time is when landing point is: 8-2.1 = 5.9 mi from the cabin
Then it will take about 1.76 hrs to get to the cabin
:
Check this
find the rowing distance: sqrt%283%5E2%2B2.1%5E2%29 = 3.66 mi
then 3.66/4 = .915 hrs rowing
find the jogging distance: 8 - 2.1 = 5.9 mi
then 5.9/7 = .843 hrs jogging
Total time: .915 + .843 = 1.76 hrs min time to the cabin