# Name:Date:Optimization problemsPeriod:A man launches his boat from a point on the bank of a straight, rectangular lake.The man can row at the rate of 6 mi/hr and can run at 8 mi/hr. He wants to reach apoint 8 miles downstream and on the other bank (point B). The river is 3 mileswide. Assume the speed of the water is negligible. He could row directly across thelake (point D) and run down the bank or he could row directly to the point B acrossthe lake or he could row to a point somewhere in the middle (C) and run the rest ofthe way. How can he reach point B in the least time?Guen: Yow mynrTm/hy2PAD 232C4A ClatexATotal time3-X(2x)- o).( 2x)-292 (0Ca(Cx9)&-13-x)10)3 10(x-92436बहका: हेCo(x2-4)Find the area of the largest rectangle that can be inscribed in an equilateral triangleof side length L assuming one side of the rectangle lies on the base of the triangle.Avea bhathPeriod:unlean aua- 14 the radius of its base. SupposeA

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I'm not sure what do to next at this point. Do I have to find the equation for how long it takes for the man to row from A to C and then to B?

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Step 1

As in the given question we can see the diagram, with the help of the diagram we find the Objective function: The point C lies in the idle of B and D.

So, the distance between C and D is (CD) = x

Then distance between B and C is (BC) = 8 – (CD) = 8 – x

Now we need to find AC

Here we use Pythagorean theorem:

Step 2

Now the total time as a function of x is:

Step 3

Now to maximize the function We find the critical points thr...

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