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MathWe need a geometric progression such that the common ration between terms is 1/3 We wil assume that: a1, a2, a3, a4, a5 are terms of a geometric progression such that a1 is the first term and r=...

MathGiven: a = 2b ............(1) b + c = 3 ................(2) c  a = 6 ................(3). We need to determine the values for a, b , and c. We have a system of three equation and three...

Mathl 2a5 l  l 3a +3 l = 0 First we will move l 3a+3l to the right side. ==> l 2a5 l = l 3a + 3 l Now we have 4 cases. Case(1): (2a5) = ( 3a+ 3) We will combine like terms. ==> a = 8 ==>...

Mathlog x^3 log 10x = log 10^5 We will use logarithm properties to solve for x. We know that log a  log b = log a/b ==> log x^3  log 10x = log x^3/10x = log x^2/10 ==> log x^2/10 = log 10^5...

MathFactor the following:a. ( 2x^2  4x 30. First we notice that 2 is common factor for all terms. Then we will factor 2. ==> 2 (x^2  2x  15) Now we will factor between brackets. ==> 2( x 5)...

MathI am assuming that 4, 7 , and 9 are the length of the sides of the triangle. We are given the sides of the triangle. Then, we will use the following formula. Area (A) = sqrt S( Sa) ( S b) (Sc)...

MathGiven the point P(0, a) is midpoint of A(2,5) and B(b, 3). We need to determine a and b. We will use the midpoint formula to determine a and b. We know that the midpoint formula is: XP = ( xA+ xB)...

MathGiven x = 2. Let E (x) = x^2/3 ( x^120/ x^110) * x^7 Then, we need to calculate E(2). First, let us simplify the expression. x^2/3 ( x^120 / x^110) * x^7 Let us simplify between the brackets. (...

MathGiven that a cars travels at 64 mph speed. Let the speed be S = 64 mph. We need to determine the distance that the car will travel after 15o minutes assuming that the average speed is 64 mph. Let...

MathGiven the parabola f(x) = 2x 3x^2 + 5. We need to determine the extreme values of the function. We know that the extreme values are the values of f(x) such that the function has maximum or minimum...

MathGiven the equations: x^2 + y^2 = 29..............(1) x+ y = 7.......................(2) We have a system of two equations and two variables. Then, we can use the substitution or the elimination...

MathWe need to determine the area of the circle whose circumference is 34pi. We know that the area of the circle is: a = r^2 n pi where r is the radius. Then we need to determine the radius. Given...

MathTo find the area of the rectangle, first we need to determine the length and the width. Let the length be L and the width be W. Given that the length = 3 times the width. Then we will write: L =...

MathThe midpoint rule basically estimates the area under a curve by sampling it a n1 points, and using the rectangle with height f(Xn) and width (ba)/n at each point to give the area under that...

Mathsec^2  2secx cosx + cos^2x = tan^2x  sin^2x (sec x  cos x)^2 = (sec ^2 x  1) sin^2 x Remember, sec^2x  1 = tan^2 x : (sec x  cos x)^2 = tan^2 x * sin^2 x (sec x  cos x)^2 =( tan x * sin x...

MathLet f(x) = ax^2 + bx + c be a quadratic function. Given the roots of f(x) are: x1= 2/3 x2= 4 Then, ( x x1) and ( x x2) are factors of f(x). Then, f(x) = ( x 2/3) ( x+4) or we can write: f(x) =...

MathWe'll expand the squares using the formula: (a+b)^2 = a^2 + 2ab + b^2 (ab)^2 = a^2  2ab + b^2 (2x1)^2 = (2x)^2  2*(2x)*1 + 1^2 (2x1)^2 = 4x^2  4x + 1 (1) (x3)^2 = x^2  2*x*3 + 3^2 (x3)^2 =...

MathLet the digits of the number be x and y. So we have 10*y + x  (10*x + y) = 54 => 10y + x – 10x – y = 54 => 9y – 9x = 54 => y – x = 54 / 9 = 6 => y = 6 + x Now, x can be 1, 2...

MathWe know that n! = 1*2*3…*n. (n+1)! = 1*2*3*4…*(n+1) (n1)! = 1*2*3…*(n1) Now, (n+1)! / (n1)! = 56 => (n1)!*n*(n+1) / (n1)! = 56 => n*(n+1) = 56 => n^2 + n – 56 = 0 => n^2 +...

MathLet the number of red balls be x. So the total number of balls is x+20. Now the probability that a ball picked at random is red is 20/(x+20) As the probability is given as 4/6 => x / (x + 20) =...

Mathx^2 + xy + y^2 = 200 and x + (xy) ^ (1/2) + y = 20 Now x^2 + xy + y^2 = 200 => (x + y) ^2  2xy + xy = 200 => (x + y) ^2  xy = 200 Also, x + (xy) ^ (1/2) + y = 20 => x + y + sqrt xy = 20...

MathWe are given the function y = x/4 + 3 and have to find the inverse. => y = x/4 + 12/4 => y = (x + 12)/4 => 4y = x +12 => x = 4y – 12 interchange x and y => y = 4x – 12 Now let...

MathIt is given that y = 3^ (sin x) We take the natural logarithm of both the sides and use the relation ln a^b = b*ln a. => ln y = ln (3^sin x) => ln y = (sin x) (ln 3) Now differentiate both...

MathIt is given that the perimeter of the rectangle is 7 times its width. Let the sides of the rectangle be denoted by W and L. Perimeter = 2(W + L) = 7W 2W + 2L = 7W => 2L = 5W => L = (5/2)*W...

MathHere we have to use ratios. An actual distance of 1400 kilometers is represented as 2.8 cm on the map. The ratio of actual distance to that on the map is 1400: 2.8*10^5 => 1400/ (2.8*10^5):1...

MathThe roots of a quadratic equation ax^2 + bx + c = 0 can be represented as [–b + sqrt (b^2 – 4ac)]/2a and [–b  sqrt (b^2 – 4ac)]/2a Now if the two are equal [–b + sqrt (b^2 – 4ac)]/2a =...

MathThe total of all the angles in a triangle is equal to 180 degrees. Now we are given that in the particular triangle the largest angle is 6 times as large as the smallest and the third angle is 30...

MathLet the point be (x,y). Now the slope of the tangent to a point on any curve is the first derivative of the curve. Here y = 1 + 1/x => y' = 1/x^2 Now, the slope of a line between (x1, y1) and...

MathWe have to find the maximum volume of the cylinder. Now let the height of the required cylinder by equal to h and let its radius be equal to r. Now the radius of the base of the cone is 4 cm. And...

MathSpeed is given by the relation, speed = distance/ time. This can be written as time = distance/ speed. Now the person runs 100 m at 15 m/s. The time required is 100/15. The next 400 m are run at 10...

MathHere we have to find the slope of dy/dx at the point ((1/ sqrt 2), sqrt 2). We are given that x = cos t and y= 2 sin t. Now from x = cos t , we get dx/ dt = sin t = y/2. From y = 2 sin t we get...

MathYou want to know why x = 3/5 is a solution for 4^(5x3) = 1. Now in 4^(5x3) = 1, the solution of x is a value that if substituted for x makes the right hand side equal to the left hand side. If we...

MathWe have to solve for w given the equation 8 + 18 / w = 14. 8 + 18 / w = 14 multiply all terms by w => 8w + 18 = 14w subtract 8w from both the sides => 14w  8w = 18 => 6w = 18 divide both...

MathWe have to find all the solutions of x^4  3x^2 + 2 = 0. Now let y = x^2 x^4  3x^2 + 2 = 0 => y^2  3y + 2 = 0 => y^2  2y  y + 2 =0 => y*(y  2)  1*(y  2) =0 => (y  1)(y  2) = 0...

MathWe have to find x if 28 = 19 + 12^(x7). Now we can rewrite 28 = 19 + 12^(x7) => 28  19 = 12^(x7) => 9 = 12^(x7) The best way to proceed is to use logarithms. => log 9 = log [...

MathFirst, we'll use the negative power property of exponentials: 1/e^(x12) = e^(x12) Now, we'll rewrite the equation: e^(3x4) = e^(x12) Since the bases are matching, we'll use one to one...

MathWe'll be able to use substitution, only after changing the first equation. 2xy=5 We'll sutract 5 both sides: 2x  y  5 = 0 We'll add y both sides: y = 2x  5 (1) Now, we can substitute y by it's...

MathWe'll start by rewriting the last term of the equation: cos 2x = cos (x+x) cos 2x = cos x*cos x  sin x*sin x cos 2x = (cos x)^2  (sin x)^2 We'll substitute the term (sin x)^2 by the difference 1...

MathWe'll note the given function as y = x*cos2x Since we have to determine the first derivative of a composed function, we'll apply the chain rule and also the product rule. dy/dx = [d/dx(x)]* cos2x +...

MathTo determine the indefinite integral, we'll write the function as a sum or difference of elementary ratios. 1/(x^2  4x) = 1/x(x4) 1/x(x4) = A/x + B/(x4) 1 = A(x4) + B(x) We'll remove the...

MathGiven the car's speed (s)= 44 mph We need to determine the time ( t) that takes the car to travel 99 miles. We are given the speed ( s) and the distance ( d). Then, we can use the distance formula...

MathLet Jon's age = J Let his brother's age = B Now if we add 6 years to Jon's age ==> J + 6 Then we will divide the sum by 3 ==> ( J + 6) / 3. Then the result of the operation will result in his...

Matha^2 + b^2 = ( a+ b) ^2 Let us simplify the expression. We know that: (a+ b)^2 = a^2 + 2ab + b^2 We will substitute into the equation above. ==> a^2 + b^2 = a^2 + 2ab + b^2 Now we will subtract...

MathGiven that the length of the rectangle ( L ) = x^5/ (x+1) Also, given that the width ( w) = (x+1)/ x^3 We need to determine (x) such that the area of the rectangle is 16 square units. We know that...

MathIf x= 4 what is ( x^3/2) * ( x^100/ (x^99) Let E(x) = (x^3/2) * ( x^100/ (x^99) First we will simplify the expression. Let us review the exponent's properties. We know that: x^(a) = 1/(x^a)...

Math2/3  ( ( 3/5) Let us rewrite: 2/3 1(1)3/5) First we will start from the right to the left. We know that 1*3/5 = + 3/5 ==> 2/3  ((3/5) = 2/3 1 ( + 3/5) Also, 1*+3/5 = 3/5...

Math61 = 7*10^(t) To find t's value, we need to isolate (t) on one side by itself. Then we will divide by 7. ==> 61/ 7 = 7*10^(t) / 7 ==> 61/7 = 10^(t) We know that: a^b = 1/a^b ==> 10^t...

MathTo find the equation of the line that is perpendicular to 9x8y = 8 and passes through ( 6,2). We know an equation of a line perpendicular to ax+by+c = 0 is of the form bxay +k = 0. Therefore a...

Math[( x^3 + 3x^2)/(x^716x^5)] /[ ( x^2+4x + 3)/(x^23x4)] Let us take each term and simplify then we will rewrite. (x^3 + 3x^2 ) Factor x^2. ==> ( x^3 + 3x^2) = x^2( x+ 3).........(1) Now we...

Matha. 25ab10ax  20b + 8x First we will rearrange terms. 25ab  20b + 8x  10ax Now we will factor 5b from the first 2 terms. ==> 5b(5a4) + 8x  10ax Now we will factor 2x from the last two...