Expert: Scotto - 4/21/2008

Question
QUESTION: a swimming pool whose bottom has area 5000/3.14 square feet is to be built in the form of a rectangle with semicircles attached at two opposite ends of rectangle. give dimensions of pool having minimum perimeter

ANSWER: What you would have for area would be a circle (the two ends when joined) and a rectangle.  Find the circles radius r and the area is pi*r^2.  Find the rectangle area, length times width, and add this on.

Also note if you sum these two area, you get the area of the pool, which can be set to the one given.

The perimeter would be the same as twice the rectangles length plus the circumference of the circle.  This is because the width of the rectangle is in the pool and there is a half circle on the outside at either end.

Using the perimeter equation, the width of the ractangle can be solved for in terms of the radius of the circle.  This can be put into the original area equation so that now you have a formula with only one variable.  Take the derivative with respect to that variable, set it equal to zero, and the answer should be found.

Once this has been done, it would be a good idea to check your perimeter equation to make sure it looks right.  This can be done by actually increasing and decreasing the value you get for the variable you solved for and then using that value to get the other.  Once you have both, the perimeter can be found.

Note that the width of the pool is the same as twice the radius of the circle.

Hope you can do it now.  If you can't or have additional questions, feel free to write and ask.  If you need to, read through the answer as each part is done to grasp all that is here.  Have a great day and take your time if necessary!


---------- FOLLOW-UP ----------

QUESTION: im sorry im still having trouble. can you explain it in more detail.

Answer
Let the circle have radius r.  The area of both of the ends (the two semicircles) is pi*r^2.

The area of the rectangle is 2r*w, where w is the width.

Add these two together, and you get 50,000/pi.

Using this, you can solve for w in terms of r since
pi*r^2 + 2rw = 50,000/pi.

The perimeter is either side of the rectange in the middle plus the edge of two semi-circles at the end.  The sum of two sides is 2w and the sum of the two perimeters is 2pi*r (since they are one circle when joined together).

You can us the solution for w in terms of r and put that into the perimeter equation.  Once this has been done, take the derivative in terms of r and set it equal to zero, then solve for r.

Read through and understand each paragraph.  You can write back and tell me which paragraph is not being understood yet.

Hope this works for you and thanks for the question.