Karl's Calculus Forum: Summer of 2004
© 2004 by Karl Hahn
© 2004 by Karl Hahn
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Reply to Kristy: This question works much the same as Dan's below, but is trickier because of the math involved. The obvious first problem is to understand the geometry, then to get your formulas straight, and last to come up with your answer. The window is made of two parts: a rectangle (which you can picture in your mind as being tall and narrow, not short and wide) and a half circle sitting on the top. You're only given one piece of informaion, the perimeter is 30. Step 1: Derive the perimeter function. What's the perimeter? Well, the rectangle has a width 'x'. This is the 'bottom' of the window. There are 2 sides, call them 'h' (for height). It has a rounded top half-circle. In order to get the perimeter (the distance around the whole window), you'll need to add up all these parts. What's the distance around the half-circle? What's the formula for the circumference of a full circle? Does P = Pi*d = 2*Pi*r sound familiar? The circumference of a half-circle is, well, half of that. Put it all together: P = x + 2*h + 1/2*Pi*x Step 2: Derive the area function. What's the area? Again, there's a rectangle on the bottom, and a half-circle on the top. The rectangle has width 'x' and height 'h', so it's area is A = x*h. The half-circle has radius x/2, so the area is A = 1/2*Pi*(x/2)2. Add these together for the full area. Step 3: Get area in terms of only the width 'x'. The problem with the area formula from Step 2 is it still has 'h' in it. How do you get rid of it? Where else do you have an 'h' floating around? Look at your perimeter formula. It has 'h' in it. Can you solve for 'h' in the perimeter formula so you can 'plug' it into the area formula? Yes! You know the perimeter is 30, so set your formula from Step 1 equal to 30, solve for 'h', and substitute this value into the area formula from Step 2. You're done! For practice (since I've seen this problem phrased this exact same way in textbooks), you might want to try and solve this question: For the given Norman window, what radius will result in the maximum possible area of window opening? (Hint: It's between 8 and 9) If you're studying max/min problems, you're bound to see this problem. Jay Lakeland, FL USA - Wednesday, August 25, 2004 at 03:29:30 (EDT) |
Dan:
First, think about the formulas for the Volume of a cube and the Surface Area of a cube. Recall the following two formulas (which hopefully you remember!):
V = s3Now, they asked for SA in terms of Volume. What's common to both of those formulas? 's', the length of a side. What you need to do is solve the Volume formula in terms of the side length 's', and then use this result in the SA formula. Once you do that, you should notice that 's' is replaced in the SA formula by V. A hint to you: be careful with your exponents! Jay Jason Karol Lakeland, FL USA - Wednesday, August 25, 2004 at 02:35:03 (EDT) |
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Hi can anyone help me with this problem.
A Norman window has the sahpe of a rectangle surmounted by a semicirle. If the perimeter of the window is 30ft. express the area A of the window as a function of the width X of the window.
PLEASE anyone help I greatly appreciate it so much.
Kristy Kristy Orlando, FL USA - Tuesday, August 24, 2004 at 17:34:53 (EDT) |
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I need help with a calc problem can anyone help me the prob is.
Express the surface area S of a cube as a function of its volume V.
Thxs Dan NC USA - Tuesday, August 24, 2004 at 17:11:43 (EDT) |
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I need help from anybody, the last time I took calculus I was 3 years ago and I need to pass calculusII for my major which is chemistry. I'm having trouble if the integrand of the definite integral is a difference of two functions(f(x),
g(x)) to sketch the graph of each function and the shade region whose area is represented by the integral.
example:
( 4(2^x/3) - x/6 )dx
b=6 a=0
chris millbury, ma USA - Tuesday, July 13, 2004 at 16:50:56 (EDT) |
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Having trouble trying to find the solution to this problem, any ideas would help thanks.
lim(x->0) [1(3+x)]-(1/3)/x Jason Smith Binghamton, NY USA - Sunday, July 11, 2004 at 15:08:18 (EDT) |
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