A manufacturer of corrugated metal roofing wants to produce panels that are u = 48 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave with equation y = sin(πx/12).
Find the width w of a flat metal sheet that is needed to make a 48-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)
Thank you Steve. I can't see to find anyway to do this question on paper. My research show that I need an integrated computer to solve this hard one but I don't know what it is.
Nor am I. I suppose if you can add special functions to your repertoire, such as Gamma, Beta, Erf, etc. Then you can just use them in your analysis. I mean, we use stuff like sin(x) all the time, but it's a pretty nasty animal that you have to look up in a table to really understand. Or, for that matter, ln(x), which is defined as ln(x)=∫[1,x] 1/x dx.
So, good luck along those lines, but you won't be able to do this one on paper using elementary functions.
Well, well, well, if it isn't my favorite topic: geometry mixed with a little bit of sine waves. Now, let's take a look at this problem and see how we can find that elusive width.
First things first, we need to determine the equation of the curve formed by the sine wave. According to the problem, the equation is y = sin(πx/12). Don't worry, we won't be singing any songs here.
Next, we want to find the width w of the flat metal sheet. In other words, we want to find the distance between two consecutive peaks of the sine curve. To do that, we need to find the x-coordinates where the y-values are at their maximum and then subtract them.
Now, we know that the maximum value of the sine function is 1. Therefore, we need to find the x-values where y = 1.
So, let's set y = 1 and solve for x:
1 = sin(πx/12)
Now, I could try to solve this equation for you, but I think you can handle it. Just grab your calculator and evaluate it to find the x-values where y = 1. Once you have those x-values, subtract them to find the width w.
And there you have it, my friend! The width w of the flat metal sheet needed to make a 48-inch panel.
To find the width of the flat metal sheet needed to make a 48-inch panel, we need to integrate the width of the sine wave over the desired width of the panel.
The width of the panel is given as u = 48 inches. The equation of the sine wave is y = sin(πx/12). We need to find the width of the flat metal sheet, which is denoted by w.
The integral to evaluate is:
∫[0, w] sin(πx/12) dx
To evaluate this integral, we can use a calculator or software that can handle symbolic calculations. Here's a step-by-step process to evaluate the integral using a calculator:
1. First, convert the width of the panel from inches to the same unit as 'x' in the equation. The equation has x/12, so we need to convert 48 inches to the same unit.
Since there are 12 inches in a foot, we have 48/12 = 4 feet.
2. Substitute the limits of integration into the integral:
∫[0, w] sin(πx/12) dx where the upper limit is 4 feet.
3. Using your calculator, input the integral with the limits:
∫[0, w] sin(πx/12) dx, where the upper limit is 4
4. Evaluate the integral using the calculator's integration function. Make sure to set the calculator to use appropriate units for angles, either degrees or radians, depending on how the sine function is defined in your calculator.
The result will be the width of the flat metal sheet needed to make a 48-inch panel.
Note: The given equation assumes x is in the same unit as the width of the panel. Make sure your calculations are consistent with the given information.
I assume that the center of the wavy sheet follows the sine curve. Otherwise you have to worry about two curves, for both top and bottom.
Now, he period of sin(πx/12) is (2π)/(π/12)=24
∫[0,48] √[1+(√π/12 cos(π/12 x))^2] dx
Still stuck with that elliptic integral, I see. There are lots of online integration sites which can evaluate it for you. wolframalpha gets 48.2607
http://www.wolframalpha.com/input/?i=%E2%88%AB%5B0,48%5D+%E2%88%9A%5B1%2B(%E2%88%9A%CF%80%2F12+cos(%CF%80%2F12+x))%5E2%5D+dx