
Math
Pi" is a letter of the Greek alphabet: `pi` . In Mathematics, it universally denotes a number which is defined as the ratio of the circumference of a circle to its diameter. This number is the same...

Math
Different people will find different topics "interesting", but here are some ideas: (1) Biographies: There are a large number of interesting people who have contributed to mathematics. There are...

Math
Hello! To factor a polynomial it is useful to find its roots. If a polynomial `P(x)` has a root `a,` then it may be expressed as `P(x) = (x  a)Q(x),` where `Q(x)` is a polynomial of the degree one...

Math
A fraction is made up of two parts, the numerator and the denominator. The numerator is the upper number of the fraction, while the denominator is its lower part. For example, in `\frac{2}{5}` ` `...

Math
In research there are several methods used for data collection, which fall into two categories: primary data and secondary data. The difference between both are explained below: Primary data:...

Math
The sequence you've posted seems to be following a pattern based on ones and threes in sets of two. With each new set, the amount of times the number is increased by one increases, while the amount...

Math
Hello! Consider the pairwise differences between the terms of this sequence. They are 16  15 = 1, 19  16 = 3, 20  19 =1, 23  20 = 3. So we can suppose that the next difference will be 1, then...

Math
By using the fact that a circle also has 360 degrees. You can measure the angles around the center. Each one will be approximately 51.43 degrees.

Math
Let us say the coordinates of the point closest to (0,3) are (a,b). The distance (L) between these two points can be given as; `L = sqrt((0a)^2+(3b)^2)` By getting `L^2` `L^2 = a^2+(3+b)^2`...

Math
Hello! Probably you mean "tangent of (x plus `pi / 2`)", not "(tangent of x) plus `pi / 2`", i.e. `tan(x + pi / 2) =  cot (pi / 2).` It is correct. The simplest way to verify it is to recall that...

Math
We have to expand the formula ` ev=log_2(\frac{f^2}{t})` using properties of logarithms. We know that, ` log(\frac{a}{b})=log(a)log(b) ` and, `log(a^2)=2log(a) ` Using the above we can write, `...

Math
We are given a rectangular yard with area of 24 square meters, and we are asked to find the possible perimeters. Since the yard is a rectangle (a parallelogram with right angles), we can describe...

Math
Hello! We know the formula `sin(u + v) = sin(u)cos(v) + cos(u)sin(v),` but it is not enough because we a given only `sin(u)` and `cos(v),` but neither `cos(u)` nor `sin(v).` But we can find them...

Math
We are asked to find `lim_(x>infty) ((3x)/(e^(2x)+7x^2))^(1/x)` : First, use the fact that the limit of a product is the product of limits to get: `=lim_(x>infty)3^(1/x) *...

Math
Hello! It is a relatively simple task. The only formula we need to solve this is the formula of cosine of a sum of two angles: `cos( u + v ) = cos( u ) * cos( v )  sin( u )*sin( v ).` This formula...

Math
How to solve 3xy=6: To solve an equation is to find all of the values for the variables that make the equality a true statement. A linear equation in one variable will have either no solutions,...

Math
Table 4 in this study presents the results of an ANOVA analysis looking at only females. This table compares the differences in victimization scores and negative outcomes (including poor academic...

Math
Hello! Denote the elevation angle as `alpha(t)` (in radians, `t` is in hours). Denote the known speed as `V` and the (unknown) horizontal distance between the airplane and the kangaroo that...

Math
The definition of derivative of a function `f(x) ` is `f'(x) = lim_(h >0) (f(x+h)  f(x))/h` (the limit of the quotient of the difference between f(x + h) and f(x) and h, as h approaches...

Math
`2x4(3x+6)=6(2x+1)4` Remove bracket by multiplying; `2x4xx3x24 = 6xx2x64` `2x12x24 = 12x64` `10x24 = 12x10` Take all x parts to one side, and put the numbers on the other side....

Math
This is not a very clear question, but it seems that you need to show that the complex solutions of the equation of the form `x^3 + a^3 = 0` or `x^3  a^3 = 0` will contain a radical of 3...

Math
We are asked to find the derivative of cot(log(x)): (1) Assuming that log(x) is the common logarithm (logarithm base 10) of x: Note that the derivative of cot(u) (where u is a differentiable...

Math
We are asked to find an antiderivative of `(2x+5)^6 ` : This is essentially equivalent to evaluating the indefinite integral `int (2x+5)^6 ` Now `int u^n du=1/(n+1) u^(n+1) ` with u a...

Math
Given that a funnel is in the shape of a cone with radius = r= 5 inch height = h= 20 inch We know that the volume of the cone is given by the formula: `Volume=V=\frac{1}{3}\pi r^2 h` i.e....

Math
Hello! We can find the rate at which sand is leaking in volume per time. It is natural to measure time in minutes and volume in cubical inches for this problem. Denote the radius of the cone (and...

Math
There is a story behind this problem. It is said that a famous mathematician by the name of Carl Gauss was punished for poor behavior in class, so he was told to stay after school. As punishment,...

Math
Some of the early computers that used decimal system (either exclusively or in combination with binary) include ENIAC, UNIVAC I and II, and several IBM computers. Some processors today use binary...

Math
The question is a little vague. Ten huskies are to be selected to pull a dogsled. The number of ways in which this can be done is dependent on the total number of huskies from which these ten are...

Math
Hello! The most straightforward way is to use Cramer's Rule. The main determinant of this system is `D = [a,b],[b,a] = a^2b^2 = (ab)(a+b).` If it is nonzero, the system has the only solution....

Math
Hello! I suppose the equation is `1.25+5.25*k` . It may also be `1.25*k+5.25.` a. Then we see that the number `1.25` (the free term) is in dollars (or other currency) and it represents the fixed...

Math
The bell curve essentially "sums up" all the possibilities that can occur given a particular experiment. Especially, in many cases, as the number of times the experiment is repeated, and all...

Math
a) Denote the number of hours it takes Ashlee to repair a DVD by h. Then, the amount of money M Ashlee would charge to repair a DVD, in dollars would be M = 25 + 20h ($25 to inspect and $20 per...

Math
(a) We have to solve the initial value problem given by: `x'=10y` `y'=10x` with initial conditions: `x(0)=3` `y(0)=4` Now we can write the above problem in matrix form as shown:...

Math
According to the variations of parameters method, the formula for the particular solution of the equation of the form `y''+q(x)y' + r(x) y = g(x)` is `Y_P = y_1 int y_2(x)g(x)/(W(y_1, y_2))...

Math
The divergence theorem states that `int_S (F*hatn) dS = int_V (grad*F) dV,` where `S` is a closed surface, `V` is the volume inside it and `F` is a good enough vector field defined inside `S` and...

Math
This function is odd, thus its Fourier expansions contains only `sin(nx)` terms, i.e. `a_n=0, ngt=0.` This expansion has the form `sum_(n=1)^(oo) b_n sin(nx)` where `b_n = 1/pi int_(pi)^(pi)...

Math
We are asked to graph the polar function `r=sqrt(theta), 0<= theta <= 2pi ` , along with its vertical and horizontal tangents. We can graph by plotting points: `theta: ` r:0...

Math
We are given the following coordinate pairs: (17,36), (27,25), (37,20), (47,12), (57,10), (67,7), and (77,5), where the xcoordinate represents the age in years and the ycoordinate the percent of...

Math
The method of disk or washers is based on breaking up the solid obtained by the revolution of the curve into either thin disks or thin washers (rings), and integrating their area to obtain the...

Math
The method of shells is based on breaking up the solid obtained by the revolution of the curve into thin cylindrical shells, and integrating the surface area of the shells to obtain the volume of...

Math
i, j, k are unit vectors. r(u,v)=(b+a*cos(u))*cos(v) i + (b+a*cos(u))*sin(v) j + a*sin(u) k 0<=u,v<=2pi and a=1/4 and b=1 The surface area is: A=double integral of magnitude of (r_u x r_v)...

Math
Please look at the pictures attached.

Math
Denote the given points: First, find the unit vector orthogonal to (for a plane it is a constant vector). An orthogonal vector is Its unit vector is a) Therefore the equation of the plane is or b)...

Math
For each `x,y` the point `(x, y)` is the point where a vector starts and F(x, y) is a pair of its components. For `x=y=0` we have `F(x,y)=vec0` and we cannot draw it with an arrow because it has no...

Math
First octant means `xgt=0, ygt=0, zgt=0.` The formula `y=27/8 x^3` may be written as `y=(3/2 x)^3.` We need the (simple) inverse formulas, too: `x=2/3 root(3)(y), z=1y/2.` Because `zgt=0` we see...

Math
Hello! As I understand, `r = sqrt(x^2 + y^2).` The volume is the integral by dx, dy and dz of the function that is constantly equal to `1.` To set up the integral we need to understand the limits...

Math
How can I find the probability (in fractions)? S={3,5,6,8,9,12,13,14,15,16} A={3,12,5,13}...
I am assuming that we are selecting one item from a sample space of s={3,5,6,8,9,12,13,14,15,16}. We have event spaces: A={3,5,12,13} B={3,6,14,15} C={5,8,9,12,16} We can compute the probabilities...

Math
In the attached image, the boxplot for Region C is both skewed and has an outlier. In statistics, a boxplot is a way to show numerical data in their quartiles. A boxplot consists of a rectangle...

Math
Hello! Although the range is from `1` to `6,` the mean and the distances from the mean are computed using the vertical values (those below `0.2`). They are relative frequencies (probabilities) of...

Math
We are asked to graph `f(x)=(x1)^3(x+4)^2(x2) ` : First we will use only techniques from Algebra. There are a few key attributes of graphs of polynomials we are interested in: end behavior (how...