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Differentiation Calculus Problems

I'm fairly new to Calculus. Please help me work out the following three questions:-

1.) A swimmer is at a point 500 m from the closest point on a straight shoreline. She needs to reach a cottage located 1800 m down shore from the closest point. If she swims 4m/s and she walks at 6m/s, how far from the cottage should she come ashore so as to arrive the cottage in the shortest point?

2.)A window consists of an open rectangle topped by a semicircle and is to have a perimeter of 288 inches. Find the radius of the semicircle that will maximize the area of the window.

3.)A company determines that its profit equation(in millions of dollars) is given by P=x(cube)-48x(square)+720x-1000, where x is the number of thousands ofunits of software sold and 0<=x<=40. Optimize the manufacturer's profit.

Thanx in advance.
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Jaspworld
Asked:
Jaspworld
1 Solution
 
ozoCommented:
Is this homework?
How far have you gotten?
What part of the questions are you having difficulty with?
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KelvinYCommented:
Hi Jaspworld,

You already have the equation to differentiate for 3). For 1) and 2) start by writing down the equations that determine the distance to travel and the area of the window. Then differentiate them and set the result = 0. Remember that a function has a maximum or minimum where the derivative is zero. Try this by yourself and let us know where you are having problems.

Regards
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aburrCommented:
If you have graphing software, you can graph the equation in 3 to select the correct answer from the several you will get by differentiating the equation. If you do not have graphing software sketch the curve anyway. It will be good proactice for real world problems.
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Harisha M GCommented:
Hi aburr,
2) Answer is 40.32714 m

Bye
---
Harish
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Harisha M GCommented:
Jaspworld
    This is done by partial differentiation methods... If this is not a HW, then I will post the solution
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Harisha M GCommented:
Jaspworld,
3) Answer is 20 or 12
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JaspworldAuthor Commented:
No!! This in not H.W.

I already know the answers. I just don't know how to get there.

Answers: 1.) 1350m  2.) 40 in  3.)$4,456 million

For 2.) I worked out an equation for area and perimeter but I just can't work them out.

For3.) I get an answer of 40 million $ and for 1.) I cant even figure out the equations.

And once again, this is not homework.
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TalmashCommented:
1) I'll guide you a little with this nice question :
   swimmer will swim to the shore NOT in the closest (500 meters) path, call it 'X',
   find how much left for her to WALK (1800 - sqrt(....)) = Y, Y is the answer of this question.

   now, write the TIME to reach the cottage : X/(speed at water) +  Y/(speed at earth) = T(X)

   now, just diffrentiate d(Tx)/d(x) .
   solve T'(x) = 0 , and find the 'x' that solve, go back to the definition  of Y , and find out what is Y.

   hint, you should get a number that is smaller then 1800, but not smaller then 1000.

tal
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XxavierCommented:
for Q1 I recommend walking along the shore till you get to the bridege and then use that.
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Harisha M GCommented:
I have the right answer then...

2) The total area  = area of rectangle + area of semicircle
    Let the radius be x => Base of rectangle = 2x, Let height of rectangle be y

=> Area = 2xy + PI x² /2
   
Perimeter = 3 sides of rectangle + semicircle
=> Perimeter = 2x + 2y + PI x

Now, by Lagrange's formula,
F = 2xy + PI x² /2 + L(2x + 2y + PI x)

Differentiating partially w.r.t x & y seperately and equating to zero,
2y + PI x + L(2 + PI) = 0
2x + 0 + L(2) = 0

           - (2y + PI x)              -2x
=> L =  ________      =      ____
              (2 + PI)                  2

2y + PI x = 2x + PI x

  2y = 2x

 x = y

Now, Given, perimeter = 288
=> 2x + 2y + PI x = 288 => 2x + 2x + PI x = 288

(4 + PI) x = 288

x = 40.32714
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Harisha M GCommented:
I guess you have   + 1000  in the equation

The value of the equation at 12 will be 4456

12³ - 48*12²+ 720 * 12 + 1000
=4456

x³ - 48x² + 720x + 1000
Differentiating,
3x² - 96x + 720 = 0
/3
x² - 32x + 240 = 0
Solving this QE,

x = (32±sqrt(32² - 4*1*240)) / 2

= (32 ± sqrt(64)) / 2

= (16 ± 8) /2

= 12 or 4

At these points, the value of the equation becomes (with +1000)
4456 and 3176 respectively.
So, maximum is 4456

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