
Math
The sides of the triangular park are of 80 m , 100 m and 120 m. The area of the park can be calculated using Heron’s formula which gives the area as sqrt [s*(s – a)(s – b)(s – c)] where a,...

Math
f(x) = (x2)/(2x3) To find the derivative, we will use the quotient rule. Let f(x) = u/v such that: u= x2 ==> u' = 1 v= (2x3) ==> v' = 2 Then we know that: f'(x) = [ u'*v  u*v']/ v^2...

Math
Given the point (a,2) and the point (3a,5). We need to find the distance in terms of a. We will use the distance between two points formula to calculate. ==> We know that : D = sqrt[ (x1x2)^2...

Math
Let the distance both cars travels before they meet is D. The time need to for car 1 is T1 The car needed for car 2 is T2 But car 2 leaves 30 minutes ( 0.5 h) after car 1 ==> T2 = T1  0.5...

Math
2 5*l 2x3 l < 7 First we will need to isolate the absolute values on the left side. Let us subtract 2 from both sides. ==> 5*l2x3l < 5 Now we will divide by 5 and reverse the...

Math
Given the right angle triangle at B is ABC such that: AB = 6 BC= 7 Then the hypotenuse is AC Now we will calculate the length of the hypotenuse. ==> AC^2 = AB^2 + BC^2 ==> AC^2 = 6^2 + 7^2 =...

Math
Let the length of the rectangle be L and the width is W. Given that the length is twice the width. ==> L = 2*W ............(1) Also, given that the area of the rectangle is 98. ==> L*W = 98...

Math
We are given the equation of the line 2xy+2 = 0 We will rewrite into the slope form. ==> y= 2x+2 ..........(1) Also, given the equation of the curve g(x) = x^2 3x 7 We need to find the...

Math
Given the curve p(x) = (x2)(3x1) We need to find the integral of p(x) between 0 and 1 First we will simplify and open the brackets. ==> p(x) = 3x^2 x 6x +2 ==> p(x) = 3x^2 7x +2 Now we...

Math
Given the systems: 2a = b..........(1) a 3b = 12 ..........(2) We need to solve for a and b. We will use the substitution method to solve. We will substitue (1) into (2) ==> a  3b = 12 ==>...

Math
We have the curve g(x) = 64x^2 We need to find the maximum values of the curve. First, we will need to find the first derivative for g(x) and determine the critical values. ==> g'(x) = 8x = 0...

Math
Given that f(x) = e^(3x)  k We need to find the inverse function f^1 (x) Let f(x) = y ==> y= e^(3x)  k Now we will add k to both sides. ==? y+k = e^(3x) Now we will apply the natural...

Math
Given the system: 2xy = 17 ............(1) 3x +2y = 4 ...............(2) We need to solve for x and y. We will use the elimination method to solve. We will multiply (1) by 2 and add to (2). ==>...

Math
Given that (2x1) , 8, (3x3) are terms of an Arithmetical progression. We will assume that the common difference is (r). Then we know that: (3x3) = 8 + r ................(1) (2x1) = 8 r...

Math
Given the sides of a triangle : 3, 6, and 7 We need to find the area. We know that the area of a triangle given the sides as follow. ==> A = sqrt(s*(sa)*(sb)*(sc) such that S = perimeter/2,...

Math
(3+5)^2 / 2 First we will solve between the brackets. ==> 8^2 /2 Now we will raise to the 2nd power. ==> 64/2 = 32. ==> (3+5)^2 / 2 = 32

Math
Given the line segment AB such that the coordinate of A is ( 2,4) and the coordinates of y is (2,1) We need to find the length of the line segment AB. We will use the distance between two point...

Math
Given the curve f(x) = x^2 4 and he line y= 2x5 We need to find the intersection points between the line and the curve. The point of intersection must verify the equations for the line and the...

Math
Given the functions f(x) = (x2)(2x+3) + x^3 We need to find the first derivative f'(x). First we will simplify the functions by opening the brackets. ==> f(x) = 2x^2 + 3x  4x  6 + x^3 Now we...

Math
Let us assume that one of the integers is n. Then, we know that the next consecutive integer is n+1 Now given that the sum of both numbers is 25. Then we will rewrite into algebraic form. ==> n...

Math
Given the equation: 9 l 5x3 l = 90 We need to solve for x. First, we need to isolate the absolute value on the left side by itself. Then, we will divide by 9 both sides. ==> l 5x3 l = 10 Now...

Math
Given the equation x^2  3x = 28 We need to find x that verifies the equation. Let us rewrite the equation so the right side is 0. ==> x^2  3x 28 = 0 Now we will factor the quadratic...

Math
The circles are tangent to each other if they touch each other only at a single point. We have the equation of the circles: x^2 + y^2  6x + 1 = 0 ...(1) x^2 + y^2  2y + 8x  1 = 0 ...(2) Equating...

Math
The general equation of a circle with radius r and center (a, b) is (x  a)^2 + (y  b)^2 = r^2 We have the equation of a circle given to us to: x^2+y^24x6y=12 Let us determine the center and...

Math
The circle we have to determine has its center at (4,3) and it is touching the line 3x + y = 15. The distance of the point (4 , 3) from the line 3x + y + 15 = 0 is given by  3*4 + 3 + 15/ sqrt (...

Math
The function of the displacement of the car with respect to time is given by f(t) = 3t^3 + 19t + 2. The function for velocity is the first derivative of the function for displacement and the...

Math
We have to find the points of intersection of x^2 + 2y^2 + y  1 = 0 with the line x + 2y = 0. x + 2y = 0 => x = 2y => x= 2y Substitute this in the other equation x^2 + 2y^2 + y  1 = 0...

Math
The person earns $74500, this can be divided into the three slabs as $25000 in the first, $25000 in the second and we are left with 74500 – 50000 = 24500 in the third. The tax paid on the first...

Math
We have to integrate f(x) = x/sqrt ((x^2)+1). First, we convert the function to make integration easier by using substitution. let x^2 + 1 = y => dy / dx = 2x => x dx = dy/2 Now f(x) can be...

Math
We have to solve 27a^3 + 27a^2  18a  18 = 0 27a^3 + 27a^2  18a  18 = 0 => 27a^2(a + 1)  18(a + 1) = 0 => (27 a^2  18)(a +1) = 0 a + 1 = 0 => a = 1 27a^2  18 = 0 => a^2 = 18/ 27...

Math
We have to solve 2*cos 2x + 4*sin x = 3. We use the relation: cos 2x = 1 – 2* (sin x)^2 2*cos 2x + 4*sin x = 3 => 2* (1 – 2* (sin x)^2) + 4*sin x = 3 => 2 – 4*(sin x)^2 + 4* sin x = 3...

Math
The student wants to have $8000 at the end of 5 years by placing a certain amount in an account every year where he is able to earn 5.2 interest annually. Let the amount be X. The amount that is...

Math
As the numbers have a common difference, they form an AP. We can take the first number as a and if the common difference is d, the numbers are of the form a , a + d … , a + 7d Now we have the sum...

Math
We have to prove that Sigma (k = 2 to inf) [ ln(k1) + ln(k+1)  2 *ln k]=  ln (2) Sigma (k = 2 to inf) [ ln(k1) + ln(k+1)  2*ln k] Now for k = 2 to inf., we have ln(k1) + ln(k+1)  2*ln k +...

Math
We have the equation of the circle given by x^2 + y^2 + 4x – 8y  5 = 0 x^2 + y^2 + 4x – 8y  5 = 0 First we complete the squares for x and y x^2 + 4x + 4 + y^2 + 16 – 8y – 25 = 0 (x + 2)^2...

Math
The slope of the line between (0, 6) and (3, 8) is [( 8  6) / (3  0)] => (2 / 3) The angle between the positive xaxis and the line is the arc tan of the slope or arc tan ( 2/3) = 33.69...

Math
The general equation of the distance between the line ax + by + c = 0 and the point (x1 , y1) is given by a*x1 + b*y1 + c / sqrt ( a^2 + b^2) substituting the values we have here: D =  3*9 + 2*6...

Math
The centroid of a triangle is the point where the medians of the triangle meet. So we can find the centroid by determining the point of contact of two medians. In the given triangle, the midpoint...

Math
To solve a system of equations ax + by = c and dx + ey = f, we need to find the inverse of the matrix formed by the coefficients of the variables and multiply it with the matrix formed by the...

Math
We know that: Speed = Distance/ Time Given the speed = 350 m/s Also, given the time = 11.4 seconds. Then we will calculate the distance. ==> 350 = Distance / 11.4 ==> Distance = 350 * 11.4 =...

Math
y= (1/3) x^3  5x  4/x ==> y' = x^2  5 + 4/x^2 ==> The horizontal tangent has a lope of 0. Then we need to find the point where the slope y' is 0. ==> x^2  5 + 4/x^2 = 0 Now we will...

Math
We have to find the equation of a circle with center (0,0). The circle touches the line 5x  12y = 52. So the line is a tangent. The radius of the circle is the distance from (0,0) to the line 5x ...

Math
Given `A= ((4, 3, 2),(5,6,3),(3,5,2));B = ((a, b, c),(d,e,f),(g,h,i))`Find B so that AB = I =...
You need to multiply the matrices A and B such that: `((4,3,2),(5,6,3),(3,5,2))*((a,b,c),(d,e,f),(g,h,i)) = ((1,0,0),(0,1,0),(0,0,1))` `((4a+3d+2g, 4b+3e+2h, 4c+3f+2i),(5a+6d+3g, 5b+6e+3h,...

Math
It is given that the points (0, 2a) and (2b , 0) are the end points of a diameter. The center of the circle is the mid point between the points (0, 2a) and (2b , 0). The center is (b , a) The...

Math
The general equation of a circle with center (a, b) and radius is: (x  a)^2 + ( y b)^2 = r^2 The center of the circle in the given problem is (4 , 2). It passes through (0, 5) So the radius of...

Math
The general equation of a circle with center (a , b) and radius r is given by ( x  a)^2 + (y  b)^2 = r^2 Substituting the values given: (x  2)^2 + (y + 3)^2 = 25 => x^2 + 4  4x + y^2 + 9 +...

Math
The circle we have to determine has its center at ( a, b) and passes through the origin. The radius of the circle is the distance between the points (0,0) and (a, b) The radius is sqrt ( a^2 +...

Math
We have to determine the equation of the circle that has a radius a and touches both the axes. This is possible if the center lies on the line x = y. Also the distance of the center from the xaxis...

Math
The area of a rectangle is the product of the width and the length. Here, we are given that the length = 21 miles. Width = 3y + 1 miles The area of the rectangle is 21*(3y + 1) => 63y + 21. The...

Math
We have to find lim x>0 [(sqrt (1+ x)  1  x/2)/ x^2] Replacing x with 0, we see that the expression is of the form 0/0, or indeterminate. This allows us to use L'Hopital's Rule and derive...