







nC0 + nC1 + nC2 + .... + nCn = 2^n 
20180219 

From bristal: (QQ) Prove, nC0 + nC1 + nC2 + .... + nCn = 2^n. Answered by Penny Nom. 





The linear thermal expansion of steel 
20110503 

From Scot: Several questions on your site deal with the linear thermal expansion of steel. Such as how much will a piece of steel grow in length if it is heated. My question is similar but I would like to know if there is a different calculation to determine how much the diameter of a round bar will grow when heated. Can you tell me how I can calculate how the diameter of .500" round steel will increase for every degree of temperature change? If a bar is raised from 60 degrees F to 120 degrees F how much will the diameter change? Answered by Robert Dawson. 





The thermal expansion of steel 
20101220 

From roger: Knowing that the coefficient of thermal expansion of steel is 6.5E06
in/in/deg F. How do you calculate the loads applied as a result of the expansion? Answered by Robert Dawson. 





Thermal expansion of a steel beam 
20100625 

From chris: If a 50 ft steel beam can expand up to 4inches when heated to 1000f
How much will a 162 ft and six inches steel beam expand under the same conditions? Answered by Robert Dawson. 





0.999 ^ (500) 
20100307 

From debra: I just need to know how to solve the following problem without using a calculator: .999 ^ (500). I know the answer is .606, I just want to do it by hand since I can't use a calculator on my test. Answered by Penny Nom and Claude Tardif. 





Finding Density Given Volumetric Thermal Expansion Coefficient 
20091226 

From florence: Hi
Please help me to apply the formula for this problem. The coefficient of volumetric for gold is 4.20 X 10^5 C degrees. The density of gold is 19,300 kg/m^3 at 0.9 C degrees. What is the density of gold at 1050 degrees C.
Could you please explain how to get the solution of 18,500 kg/m^3
Thank you for your help
Florence Answered by Janice Cotcher. 





Coefficient thermal expansion of steel 
20090629 

From roshni: Coefficient thermal expansion of steel is 0.00000645/in/in/deg F if F was C(celcius) then what is the answer Answered by Robert Dawson. 





Thermal Expansion of Steel 
20090617 

From Ken: Hi there, We are rollforming steel roofsheeting in 65M lengths and the =
question of linear expansion has cropped up.I would like to know what =
the expansion rate of this sheet would be over a temperature rise of say =
40degree F.in mm per Meter or whatever the norm is. The sheet is 0.53mm =
thick and is 700mm in width,I hope this is sufficient info to enable you =
to do your calculation.Many thanks, in anticipation.
Ken Answered by Janice Cotcher. 





A bijection from (0,1)x(0,1) to (0,1) 
20080720 

From Adam: I'm trying to prove that the function that takes the open square (0,1)x(0,1) to (0,1) is a bijection (and hence a continuum).
If we take an element (x,y) of (0,1)x(0,1) and represent (x,y) as (0.x1 x2 x3 x4..., 0.y1 y2 y3 y4...) aka x1 represents the tenths digit of x, x2 represents the hundredths, etc. Then we can define a function
f((x,y)) = 0.x1 y1 x2 y2 x3 y3... However, this is not a bijection. I hypothesize this is because you'd be unable to create the number 0.1 as x=0.1 and would have to be y=0, which contradicts the open interval (0,1) defined for y. We have been told though, if we create the same function, except that we "group" 9's with their next digit into a "block"
we can create a bijection. For example, if x=0.786923 and y=0.699213, then we define x1 to x3 as normal, but x4= 92, and x5=3. For y, we define y1 as normal, but y2=992, and y3 to y4 as normal. hence f((x,y)) = 0.7 6 8 992 6 1 92 3 3.
My questions are a) is my hypothesis on why the original function is not a bijection correct? b) why does the special blocking in the new function make a bijection? Answered by Victoria West. 





Expansion on a steel boat 
20070628 

From JOHN: that is the expansion on a steel boat that is 300' long by 100' wide.
if the water temp is 72deg f. and the deck temp is at 150deg f.
I think the deck would grow in all dir. and by what dist? do I still use
0.00000645 Answered by Stephen La Rocque. 





Motorcycle expansion chamber design 
20061114 

From David: I'm interested in calculating cone information regarding motorcycle expansion chamber design for example. I guess it's called a truncated cone, from what I've read so far. If I know the center line height, small radius, and large radius of a truncated cone then, how can I calculate the angle (included angle?) the cone forms? I'd like to know the variations of the formula so I can calculate for angle, or length, or one of the diameters if I know the other two measurements. Answered by Stephen La Rocque. 





An expansion and a translation 
20060925 

From meghan:
Write the equations for f(x) = squareroot(4  (x  2)^2) after:
a) a horizontal expansion by a factor of 2
Answer: f(x) = squareroot(4  (1/2x  2)^2)
b) a horizontal translation 3 units left
Answer: f(x) = squareroot(4  (x + 1)^2)
c) the expansion in part a), then the translation in part b)
d) the translation in part b), then the expansion in part a)
I understand how to do a) and b), but I'm not sure what I'm supposed to do for the equations in a specific order (expansion, then translation vs. translation, then expansion).
Answered by Penny Nom. 





Square roots in a binomial expansion 
20060911 

From Sydney: (√x + 5)^{4} expanded using the binomial theorem Answered by Penny Nom. 





The coefficient of thermal expansion for steel 
20051014 

From Jim:
Is the following statement true?
“The coefficient of thermal expansion for steel is 0.00000645in/in/deg. Doesn't sound like much but when you run out the numbers it comes to .405504 ft/mile/deg. Still doesn't sound like much, only about 5". Then multiply by 40 degrees and you get a piece of rail that has grown by 16.22 feet in that one mile. It's not at all unusual for the rail temp to go from say, 40 deg to 80 deg on a spring or fall day. Remember that on a sunny day, the rail temp can be significantly higher than the air temp as well."
I ran the math and came up with an answer closer to 16 inches, instead of 16 feet. Which is closer to being correct?
Answered by Penny Nom. 





Digits in the decimal expansion 
20040211 

From Leslie: In the decimal expansion of 1/17 what digit is in the 1997th place? Answered by Penny Nom. 





Newton's binomial theorem 
20030830 

From William: According to page 126 of Murtha & Willard's "Statistics and Calculus" (PrenticeHall, 1973), Newton's binomial theorem can proved inductively. I suppose that was his method, which I would like to see. Answered by Penny Nom. 





The square root of 2 
20020305 

From Roger: Does two (2) have a square root or do the numbers just keep going? Are there any other numbers that behave like two when it comes to extracting the square root? Answered by Penny Nom. 





Binomial coefficients 
20000321 

From Howard Lutz: How do you find each successive numerical term in this equation y+dy=(x+dx)^{5} =x^{5}+5*x^{4}dx+10*x^{3}(dx)^{2}+10*x^^{2}(dx)^{3}+5*x(dx)^{4}+(dx)^{5} I would appreciate an explanation of the method to find the numeric coefficient in a binomial expansion Answered by Penny Nom. 

