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MathLet the length of the pen be L and the width be W. One of the sides of the pen will be the barn. As we have to maximise the area of the pen that is constructed, the barn should form one of the...

MathWe know that the surface area of a sphere is given by the formula S=4*pi*r^2, where r is the radius of the sphere. We need the rate of change of the surface area with respect to the radius when the...

MathThe identity that has to be proved is : (tan x + 1)/(tan x)  (sec x * csc x + 1)/(tan x + 1) = (cos x)/(sin x + cos x). We know that tan x = sin x / cos x , sec x = 1/cos x and csc x = 1/sin x....

MathFirst, let us find the surface area that has to be wrapped. That is the surface area of the cylinder given as 2*pi*r*h => 2*pi*(49/2)*18 When the wire is being wrapped on this surface area, the...

MathYour question is quite vague. I assume you need to find f(x) which is a linear function. Let f(x) = ax + b 2*f(12x) + 3*f(x) = 2x + 14 => 2*[a( 1  2x) + b] + 3(ax + b) = 2x + 14 => 2a ...

MathWe have to find the sum of the first 10 terms of the series 4,10,28,82,244... Looking at the series it is not an arithmetic sequence or a geometric sequence. But if we subtract 1 from each term 4 ...

MathWe have the curves x = t^2 and y = 3t^2  t. We have to find the second derivative of y with respect to x. y = 3t^2  t => y = 3*x  x^(1/2) => dy/dx = 3  (1/2)*x^(1/2) d^2y/dx^2 = 0 ...

MathThe slope of a curve y = f(x) at the point (x, y), where y is the dependent variable and x the independent variable is given as the value of the derivative of y with respect to x. We have (y/x ...

MathThe logarithm function is defined such that log(a) b = c, implies b = c^a. Here a is called the base of the logarithm. If we have the notation merely as log without a base specified it denotes the...

MathWe have to solve the following system of equations: 2x  z = 5...(1) x + 2y = 7...(2) x + y + z = 2 ...(3) (1) gives 2x  z = 5 => z = 2x  5 (2) gives x + 2y = 7 => y = (7  x)/2 Substitute...

MathWe have to find the solution of sqrt(3m  2) + 2 = m sqrt(3m  2) + 2 = m subtract 2 from both the sides => sqrt(3m  2) = m  2 square both the sides => 3m  2 = (m  2)^2 => 3m  2 = m^2...

MathThe plane P1: x + y + 3z = 2 intersects the plane P2: x  y + 2z = 0 in a line. Determine the cross products of the normal vectors of the two planes P1 and P2. < 1 1 3 > X < 1 1 2 > is...

MathThe scalar equation of a plane is a function of a vector in a plane and its normal vector. no one vector can fully describe a plane, but the scalar equation of a plane always fully describes its...

MathThe definition of the word scalar, a real number associated to an appropriate unit of measure. Vectors are the convenient to use in determining the equation of a line in space....think of it as...

MathWe have to solve the equation : (3x + 1)/4 = (4x  1)/8 + 2 (3x + 1)/4 = (4x  1)/8 + 2 Let's make the denominators of all the terms the same. => 2*(3x + 1)/8 = (4x  1)/8 + 16/8 => 6x + 2 =...

MathYou need to write the factored form of the number 595 such that: `595 = 1*5*119` Hence, the divisors of 595 are `{1;5;119;595}` You need to write the factored form of the number 252 such that:...

MathWe need to find lim x>0 [(sqrt(x+5)  sqrt(5  x))/3x]. If we substitute x = 0 in the given expression we get the indeterminate form 0/0. We can use l'Hopital's rule here and substitute the...

MathWe need to find the solution for x^3 + 5x^2 x  5 < 0. x^3 + 5x^2 x  5 < 0 => x^2( x + 5)  1(x + 5) < 0 => (x^2  1)(x + 5) < 0 => (x  1)(x + 1)(x + 5) < 0 For the...

MathWe have the function f(x) = x/(x  1)^2 and we have to find the first and the second derivatives. f(x) = x/(x  1)^2 => f(x) = x*(x  1)^2 Use the product rule to calculate the first...

MathWe have to find the minimum g(t)=t^325t+6 on the interval [2.5, 5]. First, we find the derivative of g(t). That is equated to 0 to solve for x. The value of the second derivative is positive...

MathThere can be infinite number of planes that intersect the planes P1: x + y + 3z  2 = 0 and P2: x  y + 2z = 0. The required plane should not be parallel to x + y + 3z = 2 or the vector < 1, 1,...

MathWe need to find x and y from 3x  ix  2y = 12  iy  i 3x  ix  2y = 12  iy  i => 3x  2y  ix = 12  i(y + 1) equate the real and the complex coefficients. 3x  2y = 12 x = y + 1 substitute...

MathA man buys a dozen cameras for $1800. He sells them at a profit of $36 each. Find his profit as a...The man buys a dozen or 12 cameras for $1800. They are sold at a profit of $36 each. The initial cost of each camera is $1800/12 = $150. The profit made for the investment of $150 in buying a...

MathWe have to solve: sqrt(1  sqrt (x^4  x)) = x  1 sqrt(1  sqrt (x^4  x)) = x  1 square both the sides => (1  sqrt (x^4  x)) = (x  1)^2 => (1  sqrt (x^4  x)) = (x^2 + 1  2x) =>...

MathThe area of a trapezium with parallel sides of length a and b which are separated by a distance d is given by (1/2)*(a + b)*d. Here, we have a trapezium with parallel sides of length 6 and 30 and...

MathYou need to find the expression for the area of the square inscribed in a circle in terms of the radius of the circle given as r. A square inscribed in a circle is one that has the largest sides...

MathThis is a quadratic equation of the form: ax^2 + bx + c = 0. There are two roots or zeroes of the equation given by the formula: x1 = [b + sqrt (b^2  4ac)]/2a and x2 = [b  sqrt (b^2  4ac)]/2a...

MathIf the first term of an AP is a and the common difference is d, the sum of the first n terms is (a + a + (n  1)*d)*n/2 The sum of the first 30 terms is (a + a + 29d)*(30/2) => (2a + 29d)*15 =...

MathThe length of the ramp is dependent on the angle that it forms with the horizontal. Let us denote the angle at which the ramp is inclined to the ground to be A, and the length of the inclined...

MathThe student council originally sold tickets at $11 and the number of people who attended was 400. The total revenue with a ticket price of $11 was 400*11 = $4400. According to their estimates, a $1...

MathIn the problem the ring that is threaded in the string has a mass of 0.1 kg. The string is attached at the ends A and B with the bead hanging in the middle at point R The tension is the same...

MathWe have to solve 2x^4  11x^3 + 23x^2  19x + 5 = 0 2x^4  11x^3 + 23x^2  19x + 5 = 0 => 2x^4  3x^3  8x^3 + 12x^2 + 10x^2 + x^2  15x  4x + 5 = 0 => 2x^4  3x^3 + x^2  8x^3+ 12x^2  4x +...

MathTo calculate the total amount including the principle and the interest that $6000, deposited at an annual rate of interest of 6% compounded semiannually, becomes after 3 years, we use the...

MathAs the integers we have to find are consecutive, we can equate a = x, b = x + 1, c = x +2 , d = x + 3 and e = x + 4. x^2 + (x + 1)^2 + (x + 2)^2 = (x + 3)^2 + (x + 4)^2 => x^2 + (x + 1)^2 –...

MathWe need to find the result of the sum 3^102 + 9*3^100 + 3^103/3. 3^102+9*3^100+3^103/3 => 3^102 + 3^(100+2) + 3^(103  1) => 3^102 + 3^(102) + 3^(102) => 3* 3^102 => 3^(102 + 1) =>...

MathWe need the value of k for which x+k=2x1 and 2x+4=4(x2) have the same solution. 2x + 4 = 4(x  2) => 2x + 4 = 4x  8 => 2x = 12 => x = 6 Substitute x = 6 in x + k = 2x  1 => 6 + k...

MathWe have to find t for (e^te^t)/2=3 (e^te^t)/2=3 => (e^te^t) = 3*2 let e^t = y => y  1/y = 6 => y^2  1 = 6y => y^2  6y  1 = 0 y1 = 3 + sqrt 40 / 2 => 3 + sqrt 10 y2 = 3 ...

MathWe have to find the inverse function of f(x) = sqrt(x  4) + 3. Let y = f(x) y = sqrt(x  4) + 3 => y  3 = sqrt(x  4) square both the sides => (y  3)^2 = (x  4) => (y  3)^2 + 4 = x...

MathThis is a fairly elementary question posed by you and the answer is one that you should be able to derive on your own. I have provided the same for your convenience. 5 cups of water are to be used...

MathThe wheel makes 1 7/20 rotations in one minute. That is equal to 1 + 7/20 => (20 + 7)/20 => 27/20 rotations in a minute. If it rotates for 48 minutes the number of times it rotates is...

Math[I can answer only one of your questions, so I'll simplify 5x/(x^29) + 4/(2x6), you can use the same steps for the other.] 5x/( x^29) + 4/(2x  6) change x^2  9 to (x  3)(x + 3) => 5x/(x ...

MathWe have the vertex of the parabola as (2, 3) and the directrix is y = 7. The distance of the vertex from the directrix is 3  (7) = 4. This is equal to a. The equation of a parabola with...

MathThe parabola of which we have to find the equation has a vertex at (2,3) and a focus at (4,3). The equation of the parabola is of the form (y  y0)^2 = 4a(x  x0), where(x0, y0) is the vertex and...

MathI have changed the question as you have said that the sum has to be found and an equation to be solved. (2  3x)/(x^2  9) = 4/(2x  6) => (2  3x)/(x  3)(x + 3) = 4/2*(x  3) => (2  3x)/(x...

MathThe cylindrical hot tub is being emptied at a rate 5 ft^3 / 2 minutes or (5/2) ft^3 per minute. The height of the tub is 4 feet and the radius of the tub is 3 feet. For a cylinder with a height h...

MathWe have two similar cylinders, this means the ratio of their radius is equal to the ratio of their height. The lateral area of the cylinder are 196*pi and 342*pi. The lateral area of a cylinder...

MathFor a square inscribed in a circle with radius r, the diagonal of the square is equal to the diameter of the circle. Let the side of the square be s, its diagonal by the Pythagorean Theorem is...

MathMeasures of central tendency include mean, mode and median among several others. They are important in research to analyze the data has been collected. When a research is conducted, information on...

MathThe vertex of the parabola is (0,0), the axis is the xaxis and it passes through the point (3, 6). As the axis of the parabola is the xaxis it opens sidewards. The general equation of a parabola...

MathA parabola opens upwards if its general equation is of the form x^2 = 4ay and sidewards if its general equation is of the form y^2 = 4ay. For y^2 = 4ax, the vertex is (0,0). For any other vertex...