
Math
The project lasts for 7 weeks. Initially 100 workers are employed. As work progresses every Monday an additional 60 workers are hired. By the end of the project the total number of people hired is...

Math
We'll write sin^1 x = arcsin x. To determine the antiderivative of arcsin x, we'll have to determine the indefinite integral of arcsin x. Int arcsin x dx We'll solve the integral by parts: Int udv...

Math
To integrate y = x*(sin 2x) we use integration by parts which gives us : Int [ u dv] = u*v  Int [ v du] Let u = x , du = dx dv = sin 2x => v =  cos 2x / 2 Int [ x*(sin 2x)] => x*(cos 2x /...

Math
If you know f'(x) and need to find f(x) you have to integrate f'(x). f(x) = Int [ f'(x) dx] => Int [ 8x^73x^2 dx] => (8/8)*x^8  (3/3)*x^3 + C => x^8  x^3 + C The function f(x) = x^8 ...

Math
We have to prove that (x+y)^5  (5yx^2+5xy^2)(x^2+xy+y^2) = x^5 + y^5. (x + y)^5 = x^5 + 5*x^4*y + 10*x^3*y^2 + 10x^2*y^3 + 5*x*y^4 + y^5 (5yx^2+5xy^2)(x^2+xy+y^2) = 5yx^4 + 5y^2x^3 + 5y^3x^2 +...

Math
We have to solve the system of equations: y = x  3 ...(1) x = z  2...(2) y(x+z) = 2  xz...(3) substitute x = z  2 in (1) => y = (z + 2)  3 = z  1 Substitute x = z  2 and y = z  1 in...

Math
If the line (2x  y  10 = 0) is tangent to x^2+y^24x+2y=0, the two touch each other only at one point. 2x  y  10 = 0 => y = 2x  10 Substituting in x^2+y^24x+2y=0 => x^2 + (2x  10)^2 ...

Math
Two parallel lines have equal slopes. The slope of the line 2y + 4x  8 = 0 can be found by writing it in the form y = mx + c where m is the slope. 2y + 4x  8 = 0 => 2y = 4x + 8 => y = 2x...

Math
At the points where two curves intersect the x and y coordinates are the same. Now we have y^2 = x^2  9 and y = x  1. y^2 = x^2  9 => (x  1)^2 = x^2  9 => x^2 + 1  2x = x^2  9 =>...

Math
The integral of f(x) = ln x can be found using integration by parts which gives. Int[u dv] = u*v  Int[v du] Let u = ln x, du = 1/x dv = 1, v = x Int[ ln x] = x*ln x  Int [ x/x dx] => x*ln x ...

Math
Here we need to prove that cos C / sin (90  C)  1 = 1  cos B / sin(90  B)Use the relation sin (90  x) = cos x and cos (90  x) = sin xLet's start with the left hand sidecos C / sin (90  C) ...

Math
We have the integrals A and B defined as A = Int[x*(cos x)^2 dx] and B = Int[x*(sin x)^2 dx] A + B => Int[x*(cos x)^2 dx] + Int[x*(sin x)^2 dx] => Int[x*(cos x)^2 + x*(sin x)^2 dx] =>...

Math
Supposing that you want to solve the equation cos x = sin x + 1, we'll start by saying that this equation is linear and we'll rewrite it moving the function sin x, to the left. cos x  sin x = 1...

Math
First, we'll differentiate the function to get f'(x): f'(x) = a/x Now, we'll substitute x by 1: f'(1) = a/1 But, from enunciation, f'(1) = 2 => a = 2. We'll evaluate the definite integral of...

Math
We notice that if x is located in the interval [1/e , 1], the values of ln x are negative. If x is located in the interval [1 , e], the values of ln x are positive. According to these, we'll solve...

Math
We'll recall the rules of multiplying 2 integers:  multiplying 2 positive integers, you'll get a positive integer, too: (+2)x(+3) = +6  multiplying a positive integer and a negative integer,...

Math
We need to find the value of lim x>3 [ (3  x^2)/(x  3)] Substitute x = 3, we get 6/0 We see that the denominator has a root at x = 3. Therefore the graph has a vertical asymptote at x = 3....

Math
The general equation of a circle with center at (h,k) and radius equal to r is given by (x  h)^2 + (y  k)^2 = r^2. Here, we have the endpoints of a diameter as (2, 5) and (4 , 3) The mid point...

Math
The tower casts a shadow that is 176 meters long. The height of the tower is 250 meters. We can construct a right triangle with the base equal to 176 m and the height equal to 250 m. The angle that...

Math
It is given that Luc is 8 years older than Kate. After 10 years the sum of their ages would be 88. Let Luc's age now be L. As Kate is 8 years younger that Luc her age is L  8. After 10 years Luc...

Math
If a number has to be divisible by 3^n it should have at least n number of 3 when we determine its prime factors. 10! = 10*9*8...*1 In 10! the numbers that are divisible by 3 are 9, 6 and 3 9 =...

Math
It is given that x = 2 sin x  3 and y = 2 cos x +1. We have to prove that (x+3)^2+(y1)^2=4 Now (x + 3)^2 + (y  1)^2 => (2*sin x  3 + 3)^2 + (2*cos x + 1  1)^2 => (2*sin x)^2 + (2*cos...

Math
You are only allowed to ask one question at a time. Here's the solution to your first question. The diagonal of a square and any two sides form a right triangle with the diagonal as the hypotenuse....

Math
Don't need an answer any more for this question.Thankyou anyway.

Math
The original equations given by Maxwell is: B/A = (sin 2a  sin 2b) / (sin 2a + sin 2b) To convert this to a form that does not use double angles we use the following relation: sin 2x = 2*sin x *...

Math
The answer to the question was $11,056.

Math
We are given that: (x^2 y^6)/z = 3 ...(1) (x^2 z^5)/y^2 = 27 ..(2) We need to find one value of x^2y^2z^2 Multiply (1)*(2) => [(x^2 y^6)/z]*(x^2 z^5)/y^2 = 3*27 Open the brackets and simplify...

Math
You should remember that if you multiply both sides by `x + 1` yields: `(x + 1)(x^4  x^3 + x^2  x + 1) = 0` `x^5  1 = 0 => x^5 = 1` You should notice that the equation `x^5  1 = 0` has...

Math
We have the complex numbers z1 = 27  13i and z2 = 18 + 7i z1*z2 = (27  13i)(18 + 7i) open the brackets and multiply => 27*18 + 27*7i  13*18i  13*7i^2 simplify noting that i^2 = 1 => 486 +...

Math
We'll take the equationsquare root (x1) =1 – square root(2x). Since it is impossible for a square root to have a negative radicand, we'll impose that the radicand to be positive, or zero. We'll...

Math
We have to solve 4^x+4^(x+1)+4^(x1)=1344 4^x+4^(x+1)+4^(x1)=1344 => 4^x + 4^x*4 + 4*x/4 = 1344 => 4^x(1 + 4 + 1/4) = 1344 => (4^x)(21/4) = 1344 => 4^x = 1344*4/21 => 4^x = 256...

Math
We'll rewrite the general term of the string: an = (1/n)*Sum sqrt(1 + k/n) We'll identify a function f(k/n) = sqrt(1 + k/n) an = (1/n)*Sum f(k/n) If the function is continuous and it is, then the...

Math
f(x) = ln x / sqrtx ==> u= ln x ==> du = 1/x dx ==> dv = (1/sqrtx) ==> v = 2sqrtx ==> Int u dv = u*v  INt v du = 2sqrtx*ln x  Int (2sqrtx/x) dx...

Math
Yes. If you know the derivative of a function you can identify the original function by integrating what you have. During integration constants in the original function cannot be identified as they...

Math
For the general function f(x) = sqrt (ax + b), the inverse can be computed as follows. let y = sqrt (ax + b) => y^2 = ax + b => ax = y^2  b => x = (y^2  b)/a interchange x and y => y...

Math
We'll write both bases as power of 2, using the rule of negative powers: 2^(2x+2) < 2^4(x1) Since the bases are matching, we'll use one to one property of exponentials: 2(x+1) < 4(x1)...

Math
Let us assume you know the sine of 2x. Let sin 2x = y, we need to find tan x in terms of y tan x = sin x / cos x => sin x* cos x/ (cos x)^2 => [(2*sin x * cos x)/2] / [(cos 2x + 1)/2] =>...

Math
Using slopes to prove that lines are parallel is the easiest way to o the same. Else we need to find the equation of the lines and solve them to determine any points of intersection. If no points...

Math
We'll write the integral of the given function: Int f(x)dx = Int 2xdx/(x^2+1) + Int 5dx/(x^2+1) We'll solve the first integral using substitution technique. We'll note x^2 + 1 = t. We'll...

Math
To find lim x > 5 [ ln(x4)/(x5)], if we substitute x = 5 , we get ln 1/0 or 0/0 which is an indeterminate form. We can use l'Hopital's rule here and use the derivatives of the numerator and...

Math
We have the functions f(x)=x^216 and g(x)=x+2 and we need to solve fog(x)=0 fog(x) = 0 =>f(g(x)) = 0 => f(x + 2) = 0 => (x + 2)^2  16 = 0 => (x + 2  4)(x + 2 + 4) = 0 => (x  2)(x...

Math
A fraction consists of two numbers with a horizontal line drawn between them. The number written on top is called the numerator and the number below the horizontal line is called the denominator. A...

Math
We'll use the formula that is interchanging the base and argument. log(a) b = 1/log(b) a According to this formula, we'll have: log(2x)4 = 1/log(4) 2x We'll apply the product rule of logarithms:...

Math
The vertex of a parabola is the extreme point of the function. To determine the extreme point, we'll determine the critical points that are the roots of the first derivative of the function....

Math
To determine the primitive, we'll take the antiderivative of the function f'(x). Int f'(x)dx = F(x) + C We notice that the argument of the logarithm is the expansion of a binomial raised to cube....

Math
We notice that the function f(x) = ln x is continuously and it could be differentiated, we'll apply Lagrange's rule, over a closed interval [k ; k+1]. According to Lagrange's rule, we'll have:...

Math
We'll determine first the critical points. f'(x) = 16x  1/x To determine the critical points, we'll put f'(x)=0 16x  1/x = 0 16x^2  1 = 0 Since it is a difference of 2 squares, we'll substitute...

Math
We have to find lim x>0 [(cos x  cos 7x)/x] If we substitute x with 0, we get the form 0/0 which is indeterminate, this allows us to use l'Hopital's rule and substitute the numerator and the...

Math
1 + (1/tan^2 x) = 1/(sin^2 x) We will use trigonometric identities to simplify. We know that tanx = sinx/cosx ==> 1+ (1/tan^2 x ) = 1+ (1/(sin^2x/cos^2 x) = 1+...

Math
We have to find the integral of (cos x)^3. It is not possible to get the result without using substitution though I will not use substitution of cos x. We know that cos 3x = 4(cos x)^3  3*cos x...