SOLUTION: The question is as under :
" Alight house is located at A, 2 km off-shore from the nearest point O on a straight beach and a shop is located at B. The shop is located on the beach
Algebra.Com
Question 890509: The question is as under :
" Alight house is located at A, 2 km off-shore from the nearest point O on a straight beach and a shop is located at B. The shop is located on the beach , 4 km distant from O. If the housekeeper can row at the rate of 4 km/hour and walk at the rate of 6 km/hr, where should he plan to reach the shore, so as to cover the distance to the shop in the least possible time ? "
The question is related to maxima and minima chapter in differential calculus.
Please send step by step solution of the above question.
Regards,
Khoka123
Answer by josgarithmetic(39617) (Show Source): You can put this solution on YOUR website!
A light house, a shop, and a housekeeper. Where does the housekeeper begin? From the light house? I could analyze and solve the problem if you make the description understandably precise.
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I believe I have an understanding of the description after thinking through the description.
Draw a right triangle. Vertical segment OB, showing B below O, and point A to the right of point O. Segment OB is 4, and segment OA is 2. The angle AOB is a right angle.
The housekeeper will row from the light house (?) to point P between O and B.
Let y = Distance for walking which is segment PB.
That means 4-y is length of OP.
Distance Rowing,
Distance Walking, .
Uniform Rates usually for travel can be represented RT=D, or T=D/R.
Using a function for total time, T you have total time for this housekeepers row & walk trip as .
Find the derivative, .
In one of its raw forms, this starts as:
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Equating that to 0, and then omitting the many algebra steps here, I am finding ; if this has a real solution, then I believe I made no mistakes...
I MADE A MISTAKE IN MY WORK, SOMEWHERE. MAYBE YOU WILL FIND IT AND GET THE RIGHT ANSWER.
RETRIED SOLUTION-------------------------------------------------------
The same formula for T was found.
Derivative .
Simplifying and then focusing on the numerator being 0 gave .
Two solutions for y were found, and the choice which makes sense was .
The calculations and computation to find that are omitted here, taking nearly a whole page on paper.
The housekeeper would row to a point between O and B, 0.18 km from the point B (the shop).
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