The first step to optimization problems is setting up equations that characterize the constraints under which you are working. In this problem, the total weight of the nuts and bolts must add up to 100kg.
(weight of nuts) + (weight of...
Let's call the original length of the wire LW (for length warm) and the
shorter length of the cooler wire LC (for length cold).
In cooling from 21C to 0C, the wire shortens by a*(21C - 0C)*LW, where a is the coefficient of thermal expansion...
From combinatorics, the number of ways to choose n items given k flavors is "(n + k - 1) choose (k - 1)". In this scenario, n is the number of digits, and k = 10 for the digits from 0 through 9.
For example, the number of unique...
First determine the maximum torque (T_max) that the shaft will be subjected to. The maximum torque is power (3MW) divided by angular frequency (200rpm converted to rad/s).
Next calculate the polar moment of inertia in terms of the external radius,...
For Newtonian fluids in laminar flow, internal shear stresses (tau) are related to the velocity gradient (du/dy) by the equation:
tau = mu*(du/dy)
tau is shear stress.
mu is dynamic viscosity (0.048Pas).
u is the speed of fluid...
Let's assume that the original wood beam is simply-supported and capable of holding 2690lb evenly distributed along its length. This 2690lb includes the weight of the wood. The shear and tensile strength of the wood can be estimated using the following...
Here is how the function X behaves:
[0, 0.5) --> 2
[0.5, 1] --> 3
Therefore, the inverse images are:
2 --> [0, 0.5)
3 --> [0.5, 1]
any other real number --> empty set
Now, if you wanted the inverse...
You can begin both problems by replacing 314 with its modular 7 representation:
314^163 = 6^163 mod 7
Note how powers of 6 repeat cyclically in the modular 7 number system:
6^1 = 6 mod 7
6^2 = 1 mod 7
6^3 = 6 mod...
The augmented matrix is
1 2 0 2
-1 (k-3) 3 -2
2 (k+3) (k+3) 5
Its row echelon form is
1 2 0 2
0 (k-1) 3 0
0 0 k 1
When does this system have a unique solution? Try applying the fact that Ax=b has a...
You can think of the points X, Y, Z as vectors originating from the origin. Each of these "vectors" would have a magnitude associated with it, found by using the Pythagorean Theorem.
For example, the magnitude of X would be sqrt(12^2 + 9^2) =...
To visualize how the graph moves, rewrite y = (x - 3)^2 + 4 so that it is easier to compare with y = x^2.
old: y = x^2
new: (y' - 4) = (x' - 3)^2
Now you can see that the transformation changed y to (y' - 4) and x to (x' - 3).
y = y' - 4 ---->...
The only numbers that square to one are 1 and -1.
1 = 1 x 1
1 = (-1) x (-1)
So since (tan x) squares to one, (tan x) must be 1 or -1. There are two values of x in the interval [0, pi] that make it happen.
tan (pi/4) = 1
tan (3*pi/4) = -1
Here is a graphical way to think of it.
Suppose you wanted to divide the number 1 by 0. In other words, you want to find y = 1/x when x = 0. Well, try graphing the function y = 1/x. You will notice that to the left of x = 0, the value of y approaches negative infinity...
One way to solve this problem is to use the Law of Sines: sin(A)/a = sin(B)/b = sin(C)/c. Here a, b, c are the lengths of the sides, and A, B, C are the angles across from (not touching) the sides a, b, c respectively.
Since you are given b, c, and B, you could...
The typical way to evaluate 67 x 436 is by multiplying and adding: (60 + 7) x 436. Usually this is written as:
3052 = 436 x 7
+ 26160 = 436 x 60
Here is a way to remember that the angles of a triangle sum to 180 degrees.
Imagine a squat triangle, with two very acute angles and one very obtuse angle. If this triangle is very stout, then the acute angles are ~0 and the obtuse angle is ~180 degrees.
Let's suppose 9 is a factor of the number N. That means N is 9 times some integer M.
N = 9*M
Since 9 = 3*3, we can also write N as
N = 3*3*M
That means N is 3 times some integer (3*M). So 3 is also a factor of N.