
Math
To calculate the area of the sidewalk, first we will calculate the area of the small rectangle and the large rectangle and find the difference. The small rectangle (inside sidewalk) has the...

Math
To find the area of a regular polygon we use the formula: A = s^2 * n / 4tan(pi/n) such that : s= length of the side. n= number of sides. In a hexagon, we have 6 sides. Then, we us the formula as...

Math
We will use the distance between a line and a point formula to find y. We know that : D = l ax + by + c l / sqrt(a^2 + b^2) Now we have D = 12, a = 3 b= 4 c = 5 x = 2 and y= y Now we will...

Math
Given that the volume of a cube is 214. We will use the volume formula to find the length of the side. ==> V = s^3 = 214 ==> S = (214)^1/3 = 5.98 Now we will calculate the surface area of the...

Math
Obviously 0 < 2n*n^2/ n! < 2^n*n^2/n^n, as n^n > n! This implies 0 < 2^n*n^2/n! < 2^n/n^(n2) < 2^n/n^n = (2/n)^n which approaches zero as n goes infinity. Therefore 0 < Lt...

Math
The diagonal of a prism is given by D = sqrt (l^2 + w^2 + h^2) Here the length has to be found, let it be L. The width is 2 in. The height is 18 in. And the diagonal is 21 in. 21= sqrt (L^2 + 2^2 +...

Math
The sides of the rectangular prism are 15 mm, 20 mm and 8 mm. The length of the diagonal for a rectangular prism is given by D = sqrt (w^2 + l^2 + h^2) Substituting the values we have D = sqrt...

Math
For a cube the length of the diagonal is given by sqrt(s^2 + s^2 + s^2), where s is the length of the edge. Or length of the diagonal = sqrt (3*s^2) Here the length of the diagonal is 9 ft. =>...

Math
Let the small bag of chips cost S and the large bag of chips cost L. The difference in the cost of the two is $0.9 => L = S + 0.9 Alicia bought 5 large and 3 small bags for $17.22 => 5L + 3S...

Math
The volume of the prism is given by length*width*height. Substituting the values we have here: V1 = 4*3*2 => 24 The volume of a rectangular prism is (1/3)*height*area of base Substituting the...

Math
The circumference of the circle is 3 times that of a smaller circle. The circumference of a circle is given by 2*pi*r, where r is the radius. The are of a circle is given by pi*r^2 where r is the...

Math
We have to find the possibility that neither of the marbles drawn simultaneously from the box containing 14 marbles in all of which 4 are white, 6 are red and 4 are blue turn out to be white. To...

Math
log (3x^2 2) = 2 To solve we will raise to the 10th power. ==> 10^log(3x^2 1) = 10^2 ==> (3x^2 2) = 100 Now we will add 2 to both sides. ==> 3x^2 = 102 Now we will divide by 3. ==>...

Math
5 =< 3x4 =< 12 We need to find the values of x that satisfies the inequality. The goal is to isolate x in the middle. First we will add 4 to all sides. => 45 =< 3x =< 12 + 4...

Math
f(x) = 3*ln (2x1) + 5 We need to find the inverse function f^1 (x). let y= 3*ln (2x1) + 5 We need to isolate x by itself. First we will subtract 5 from both sides. ==> y5 = 3ln (2x1) Now we...

Math
Let the length of the rectangle be L and the width is W. Given that the area is 48. ==> L*W = 48 ............(1) But we know that the length is 3 times the width. ==> L =...

Math
Let f(x) = tanx We need to find f'(x). First we know that f(x) = sinx/cosx ==> We will use the qoutient rule to find the derivative. ==> f(x)= u/v such that: u= sinx ==> u' = cosx v= cosx...

Math
We have to find the partial fractions for (2x + 3)/(x^2  7x  8) First factorize the denominator x^2  7x  8 => x^2  8x + x  8 => x(x  8) + 1(x  8) => (x + 1)(x  8) (2x + 3)/(x^2 ...

Math
Given the function : f(x) = 5lnx * (x^23x) We need to find f'(x) and f'(1). We will use the product rule to find the derivative. ==> Let f(x) = u*v such that: u= 5lnx ==> u' = 5/x v=...

Math
Given the length of the sides of a triangle are 7, 8 , and 11 We need to find the area . We will use the following formula. ==> A = sqrt(S(sa)(sb)(sc) such that s is the perimeter/2 , and a,...

Math
Given the three lines: y= 2x3 .........(1) y= 3x+5.........(2) y= x11..........(3) First we will determine the point of intersection between the lines (1) and (2). ==> 2x3 = 3x + 5 ==> x =...

Math
Given the system of equations: 3x  y= 5............(1) 5x 2y = 3...........(2) We will solve the system using the substitution method. We will rewrite equation (1). ==> y= 3x  5. Now we will...

Math
log (2x2) = log (x2) + 1 we will use logarithm properties to find the value of x. First we know that log 10 = 1 ==> log (2x2) = log (x2) + log 10 Now we know that log a + log b = log ab...

Math
To determine the quadrant we will need to find the intersection point between the lines , then we determine the location. ==> 5y3x + 1 = 0 ==> y= (3x1)/5 ............(1) ==> 2y4x + 7 =...

Math
Given that sin(a) = 3/5 We need to find cos(a). First we will use trigonometric identities to find cos(a). We know that: sin^2 a + cos^2 a = 1 ==> cos(a) = sqrt( 1 sin^2 a) ==> cos(a) =...

Math
Given the curve f(x) = x^2 and the line y= 5x6 We need to find the area between the curve and the line. First we need to find the intersection points. ==> x^2 = 5x6 ==> x^2  5x + 6 = 0...

Math
Given that the volume of a cylinder is 231. Then we know that: v = r^2 * pi * h such that r is the radius and h is the height. But we are given that the circumference of the base is 8pi. ==> C =...

Math
Given that the product of a and b is 240 Then we will rewrite: a * b = 240 .............(1) Now we know that b is 4 less than twice a. Then we rewrite: b= 2a  4..............(2) Now we will solve...

Math
Let y= v(x)f(x) = ve^x ==> y' = ve^x + v'e^x ==> y'' = ve^x + 2v'e^x + v''e^x Now we will substitute into the equation. ==> ve^x + 2v'e^x + v''e^x  2(ve^x+v'e^x) + ve^x = 0 Since we need...

Math
Given the differential equation: y''''  y''' 3y'' + 5y'  2y = 0 First we will rewrite into the auxiliary form. ==> r^4 r^3  3r^2 + 5r 2 = 0 Now we will solve by factoring. ==> (r1)^3...

Math
First we will rewrite into auxiliary equation form. ==> r^2 + 2r + 4 = 0 Now we will calculatet the roots. ==> r1= [ 2+ sqrt(416) / 2 = 1+sqrt3*i ==> r2= 1sqrt3*i Since the roots...

Math
Given the differential equation: dy/dx = (2xy^2+1) / (2x^2y) First we will cross multiply. ==> (2x^2 y ) dy = (2xy^2 + 1) dx Now we will integrate both sides. ==> Int 2x^2y dy = Int (2xy^2...

Math
We have the equation x^2 + 2x  8 = 0 to solve by completion of square. We know that (a + b)^2 = a^2 + b^2 + 2ab x^2 + 2x  8 = 0 => x^2 + 2x + 1  8 1 = 0 => (x + 1)^2  9 = 0 => (x +...

Math
To solve problems that have the expression of the form 2(x + 90), you have to open the brackets. When that is done each term inside the bracket is multiplied by the term outside. In the case of 2(x...

Math
The product of n natural numbers can only be found by multiplying the n numbers. There is no formula for that. If you have tables for the factorials of numbers you could find the value of the...

Math
We have to find the partial fractions of (2x + 1)/ (x^3  6x^2 + 13x + 42) We first need to factorize the denominator (x^3  6x^2 + 13x + 42) x^3  2x^2  4x^2  8x + 21x + 42 => x^2(x + 2)...

Math
Given that the area of the square field is 576 ft^2 Then, we will calculate the length of the sides. Let the side of the field be x. ==> A= x^2 = 576 ==> x = 24 Now we will calculate the...

Math
Given that the sum of the integers a and b is 57. ==> a + b= 57 ............(1) We are given that b is 3 less than double a. ==> Then we will rewrite : b= 2a  3 ...............(2) Now, using...

Math
f(x) = 12/(x^2 4x 12) We need to find the integral. First we will use partial fraction to simplify. ==> x^2  4x 12 = (x6)(x+2) ==> 12/(x^24x12) = A/(x6) + A/(x+2) Now we will multiply...

Math
Her income was 41,250 Her income now is 48,880 We need to find the amount of income increase in percentage. First we will calculatet he difference. ==> 48880  41250 = 7630 Now we will divide...

Math
Let the sides of a triangle be a,b, and c, and let the height be h. Then We will assume that c is the base and h is the height. Then the area is given by : A 1= (1/2) * c * h ...........(1) Now...

Math
Given the area of a circle is 12pi. Then we will calculate the radius using the circumference formula. ==> C = 2* pi *r = 12 pi ==> r = 12pi/2pi = 6 Now we will calculate the area of the...

Math
Given a= 3b and c= b2 We need to find the value of (a^2 0 3ab + 12 ). ==> Let y= a^2  3ab + 12 Now we will substitute with b and c values in terms of a. ==> Given that a = 3b ==> b= a/3...

Math
f(x)= (3x+6)/ ln x^2 We know that ln x^2 = 2*ln x ==> f(x) = (3x+6)/2ln x We will use the quotient rule to find the derivative. ==> f(x) = u/ v such that. ==> u= 3x+6 ==> u' = 3 ==>...

Math
Given the line y= 5x 3 and the curve f(x) = x^2 +3 We need to find the intersection points between the line and the curve. ==> f(x) = y ==> x^2 + 3 = 5x3 ==> Now we will combine like...

Math
First we need to find the intersection points between the curve and the line. ==> f(x) = y ==> 3x^2  4x + 2 = 2x1 ==> 3x^2  6x + 3 = 0 ==> We will divide by 3. ==> x^2  2x + 1 =...

Math
You need to use derivative definition to find what derivative of function `f(x)=sin 7x` is such that: `f'(x) = lim_(hgt0) (f(x+h)f(x))/h` `lim_(hgt0) (f(x+h)f(x))/h = lim_(hgt0) (sin...

Math
You may use the following substitution `x = 240t` such that: `x  40 = 240t  40` `3x + 2 = 3*240t + 2` Changing the variable yields: `240t(240t  40)(3*240t + 2) = 240` You need to move all...

Math
We are given that y = 02591x^2 + 6327 x  2947 and dy/dx = 0. As y = 02591x^2 + 6327 x  2947 dy/dx = 2*02591*x + 6327 2*02591*x + 6327 = 0 => 5182*x + 6327 = 0 => x = 6327/5182 => x...

Math
It is given that f(x) = log(2)(x  1) and g(x) = log(2)(5x + 17). Now f(x) + g(x) = 6 has to be solved for x. We use the property of logarithms that log a + log b = log (a*b) f(x) + g(x) = 6 =>...