Calculus Optimization Problem?

Question

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Minimum Time: A man is in a boat 2 miles from the nearest point on the coast. He is to go to a point Q, located 3 miles down the coast and 1 mile inland (see figure). He can row at 2 miles per hour and walk at 4 miles per hour. Toward what point on the coast should he row in order to reach point Q in the least time?

Please provide each step.

Thanks in advance!!!

Answer 1

distance from boat to shore = sqrt(x^2+4)
distance from shore to point Q = sqrt((3-x)^2+ 1^2)
t = d/r
t = [(x^2+4)^.5]/2 + (10-6x+x^2)^.5/4
dt/dx =.5*2x/2(x^2+4)^.5 +.5(-6+2x)/(4(10-6x+x^2)^.5) We need to set this equal 0 and solve for x to get minimum time.
x/2(x^2+4)^.5 + (-3+x)/4((10-6x+x^2)^.5) =0
x/2(x^2+4)^.5 = (3-x)/4((10-6x+x^2)^.5) =0
x^2/4(x^2+4) = (3-x)^2/16(10-6x +x^2)
4x^2(10-6x+x^2) = (3-x)^2(x^2+4)
4x^4 -24x^3+40x^2 = x^4 -6x^3+13x^2 -24x +36
3x^4 -18x^3 +27x^2 + 24x - 36 = 0
x = 1

Answer 2

a million) it rather is easy; ==> First come to a call the objective of the subject; right here it rather is paper dimensions, so as that its area would be minimum; then next comes under what constraints this must be performed? ==> nicely, right here on the chosen paper, you're leaving some margins throughout and arriving on the area for printing, this is given some mounted fee. 2) Now, we are clean what we desire; the thank you to proceed for development mathematical equations, so as that it rather is solved for paying for the top answer: 3) right here we are given the printing area as 50 squarewherein is persevering with; hence enable us to start from this awareness; ==> enable the measurement of the printing area be x in (peak clever) with the help of y in (width clever) ==> Printing area = xy = 50; == y = 50/x -------(a million) 4) there's a margin of four in each and every at suited and backside are presented; hence regular peak of the paper is "x + 8" in; further a margin of two in is supplied on the two facets; ==> regular width = y + 4 in 5) hence the area of the paper is = (x+8)(y+4) 6) Substituting for y from equation (a million), A (x+8)(50/x + 4) = 50 + 4 hundred/x + 4x + 32 7) hence the function to be minimized is: A = 80 two + 4x + 4 hundred/x 8) Differentiating this, A' = 0 + 4 - 2 hundred/x^2 = 4 - 4 hundred/x^2 9) Equating A' = 0, x^2 = one hundred; ==> x = +/- 10 in 10) yet a measurement can't be adverse, hence we evaluate in elementary terms + 10; So x = 10 in and y = 5 in 11) besides the fact that if we'd desire to be advantageous,even if this is minimum or optimal; for which we can be conscious 2nd spinoff try; So lower back differentiating, A'' = 800/x^3; at x = 10, A'' is > 0; hence it rather is minimum subsequently we end for the printing the area to be 50 squarein, under the given constraints of margins, the outer length of the paper must be 18 in with the help of 9 in; So outer area is = 162 squarein. wish you're defined; have an incredible time.

Answer 3

He has to row at least one hour. If he does that, he has to walk sqrt(10) miles, or about 0.8 hours.

If he row any longer, he will move closer along the coast, but I don't think he will be able to gain any time from the realization he will be able to walk a shorter distance. (this is sort of like the guy who wants to average 60 miles an hour on a two mile roadway, and averages 30 miles an hour for the first mile).