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Math(u+ 3)/8 = 5/(u3). To solve the equation, we need to cross multiply. ==> (u+3)*(u3) = 8*5 ==> (u+3)(u3) = 40. Now we will expand the brackets. ==> u*u + 3*u + u*3 + 3*3 = 40 ==>...

Math3x^3 + 22x^2  16x = 0 First we will factor x from all terms. ==> x*(3x^2 + 22x  16) = 0 Now we will find the roots for the quadratic equation. ==> x1= (22 + sqrt(484+4*3*16) / 2*3...

MathSolve the following system of linear equations. Present your solution as an ordered pair (x,y) 3x...3x + 4y = 11 .................(1) x  2y = 7 ....................(2) We are given a system of two equations and two variables. We will use the substitution method to find x and y values. First, we...

Math12a = 20b . To find the ratio a:b, we need to find a/b. ==> But we are given that 12a = 20b Then we will divide by 12. ==> a = 20b/12 Now we will simplify the fraction. ==> a = 5b/3. Now...

Math50/ (2t+4) = (2t+4) / 2 First we will cross multiply. ==> (2t+4)*(2t+4) = 2*50 ==> (2t+4)^2 = 100. Now we will take the root of both sides. ==> (2t+4) = +10. Then, we have two cases. Let...

MathThe ratios that Celia had with her were 4: 7: 8 and 3: 8: 13. Now for a triangle the sum of the lengths any two sides should be greater than the length of the third side. If this is not the case,...

MathThe ratio of the length of the sides of a rectangle is 5: 2, and its area is 90 square feet. Find...Let L be the length of the rectangle and w be the width. Then, we know that L > W. Given the ratio between the length and the width is 5:2 ==> L/W = 5/2 We will cross multiply. ==> 2L =...

MathThe value of x of x is in (0 , pi). sinx = 7/25 and cosx = 24/25. It is not necessary for her because the given condition itself is capable of determining this. Sinx = 7/25 and cosx = 24/25 both...

MathThe answer to this question lies in the fact that trigonometric functions are periodic in nature. This means that after regular intervals the value is repeated. For example the values of the sine...

MathI am assuming that you mean cos^2 x  3/4 because 3.4 is not valid. ==> cos^2 x  3/4 = 0 Let us solve the equation. First add 3/4 to both sides. ==> cos^2 x = 3/4 Now we will take the root...

MathGiven the inequality 3=< 2b3 =< 13. We need to find all possible b values where the inequality holds. Let us solve the same way we solve the equality. ==> 3 =< 2b3 =< 13 The goal...

MathGiven the function f(x) = 2x^2 * ln x. We need to determine the first derivative. We notice that the function f(x) is a product of two functions. Then, we will use the product rule to find the...

MathGiven the equality ( x3) = 5/(2x+4). We need to find x values that satisfies the equality. ==> (x3) = 5/(2x+4). First, we need to rewrite the equation. We will multiply by the denominator...

Mathf'(x) = 2x^2  6x + 7 find f(x) if f(0) = 8. We are given the first derivative of the function and we need to determine the function. Then we know that: f(x) = integral of f'(x). ==> f(x) =...

Mathlet the numbers be x and y. Given that the mean of the numbers is 12. Then we know that the mean = sum of the numbers/total number. ==> (x+ y) /2 = 12 Multiply by 2. ==> x+ y = 24 ==> y=...

MathGiven the complex number 3z+i = 2z i + 5. We need to find the absolute values for z. First, we need to rewrite z as a complex number into the form z= a+ bi. ==> 3z + i = 2z  i + 5 We will...

MathGiven the cylinder whose volume is 39 and the height is 4. We need to find the circumference of the base of the cylinder. We know that the volume of the cylinder is given by: V = r^2*pi * h where...

MathWe have to differentiate f(x) = 2sin(x^2)*cos(x^2). Now we know that 2 *sin x* cos x = sin 2x Therefore 2*sin(x^2)*cos(x^2) => sin (2*x^2) To find the derivative we use the chain rule. =>...

MathGiven the function f(x) = e^(3x  5). We need to find the inverse (f^1 (x) of the function f(x). Let y= e^(3x5). Now we will apply the natural logarithm for both sides. ==> ln y = ln...

MathWe have to find the equation of the circle whose circumference is 14pi and the center is (3,2). Now the circumference of a circle is given by 2*pi*r where r is the radius. Here 2*pi*r = 14*pi...

MathWe'll write the vector form of the equation of the line M1M2. r = r1 + t(r2  r1) (*), where r1 and r2 are the vectors of position of the points M1 and M2. r1 = xM1*i + yM1*j + zM1*k We'll...

MathGiven the equation of an ellipse is 4x^2 + 16y^2 = 64. We need to find the center and foci. First we need to rewrite the equation into the standard form of the ellipse. ==> x^2/a^2 + y^2 / b^2...

Math(2x+ 1)(3x2) = 55 We need to find x values that satisfies the equation. First, let us expand the brackets. ==> 2x*3x + 1*3x + 2x*2 + 1*2 = 55 ==> 6x^2 +3x  4x  2 = 55 Now we will...

MathWe have to solve 5x/(2x+2)=2x/(3x+3)+(2x+3)/(x+1) for x. Now 5x/(2x+2)=2x/(3x+3)+(2x+3)/(x+1) separate the (x+1) which is common as the denominator for all the terms => 5x/ 2*(x+1) = 2x/3(x+1) +...

MathWe have to find the roots of the polynomial 4x^316x^2+21x27. Let's equate this to 0 and find the roots. 4x^316x^2+21x27 =0 => (x 3) (4x^2  4x + 9) =0 Therefore x1 = 3 x2 = [b + sqrt (b^2...

MathP(x)=3x^310x^25x. The three roots of this equation can be found by equating 3x^310x^25x to 0. => 3x^310x^25x =0 => x ( 3x^2  10x  5) = 0 So one of the roots is 0 The other two roots...

MathWe'll write 121pi /3 = 120pi/3 + pi/3 cot (121pi /3) = 40pi + pi/3 A positive cot angle t to 121pi /3 will be calculated as follows t = 120pi/3  20(2pi) t = 120pi/3  40pi t = pi/3 So, the angle t...

MathWe have to find sin x/2, given that sinx = 1/4 and x lies between pi/2 and pi. Now cos x = 1 2 (sin x/2)^2 (cos x)^2 = 1  (sin x)^2 => (cos x)^2 = 1  (1/4)^2 => cos x = sqrt [ 15/ 16]...

MathThe line d is parallel to the plane P, if and only if the vector parallel to the line, v, and the vector perpendicular to the plane P, n, are perpendicular. Two vectors are perpendicular if and...

MathFor y = tan x, the domain is all values that x can take so that the value of y is defined. The range is all values that y can take. To calculate y = tan x, we have to calculate tan x and multiply...

MathI think there has been an error in the question posted by you, it should have been 6 tan x + 4 sin x = 0 => 6sin x / cos x + 4 sin x = 0 => 2sin x ( 3/ cos x + 2) = 0 => 2sin x = 0 or 3/...

MathTo find the derivative of a function f(x) from the first principles we have to find the value of lim h> 0 [(f(x +h) – f(x))/h] We have y = sqrt (x) So y’ = lim h> 0 [(sqrt (x +h) –...

MathThe roots of a quadratic equation ax^2 + bx + c = 0 can be expressed as [b + sqrt (b^2 – 4ac)]/2a and [b  sqrt (b^2 – 4ac)]/2a. If the two roots have to be equal b^2 – 4ac should be equal...

MathYes you can use a TI83 graphing calculater to help solve geometrical problems

MathLet the value of one of the roots be R, the other root is 2R. Now the quadratic equation can be derived by the following operation: (x  R)(x  2R) = 0 => x^2  xR  2xR + 2R^2 = 0 => x^2 ...

MathThe diagonal of a cube is given as sqrt (3L^2), where L is the length of the side. We know that the diagonal is 1. => 1 = sqrt (3L^2) => 1^2 = 3L^2 => L^2 = 1/3 => L = sqrt( 1/3) The...

MathLet the length be L and the width be W. Perimeter = 2(L+W). Area = L*W. The perimeter is given as 16. Also the length is thrice the width. This gives L = 3W 2(3W + W) = 16 => 2*4W = 16 => W =...

MathWe have to solve 5*2^t = 2^2t  6 for t. Now let 2^t = y 5*2^t = 2^2t  6 => 5*y = (2^t)^2  6 => 5y = y^2  6 => y^2  5y  6 =0 => y^2  6y + 1y  6 = 0 => y( y  6) +1(y  6) = 0...

MathNow we are given that log x^3  log10x = log 40 Use the relation log a  log b = log (a/b) log x^3  log10x = log 40 => log (x^3 /10x) = log 40 => log x^2/ 10 = log 40 find the antilog of...

MathGiven the equations: 2x= 3y 5 and y/2 = x/3 + 4. We will rewrite. 2x= 3y  5 ==> 2x  3y = 5 ............(1) y/2 = x/3 + 4 Multiply by 6. ==> 3y = 2x + 24 ==> 2x + 3y =...

MathWe know the area of the circle is given by pi*r^2. And the diameter is 2*r. Now the area of the circle is 201.1 in^2. Therefore 201.1 = pi*r^2 => r^2 = 201.1 / pi => r^2 = 64.0...

MathLet ABC be a right angle triangle such that B is the right angle. ==> AC is the hypotenues = 10 ==> A = 40 degrees. ==> Then, we conclude that the angle C = 180AB = 180  90  40 = 50...

MathGiven the center of a circle is the point (0, 13). Also, given that the area of the circle = 25*pi. We will write the equation of the circle. (xa)^2 + (yb)^2 = r^2 where (a,b) is the center and r...

MathGiven endpoints of the diameter in a circle are ( 18. 13) and ( 4, 3). We need to find the equation of the circle. We will write into the circle standard equation. ( ya)^2 + ( xb)^2 = r^2 where...

MathWe are given that ABC is a right triangle. AB is equal to 10 and the Angle A is 27 degree. Also, AC is the hypotenuse. Now cos A = adjacent side / hypotenuse => cos A = AB/ AC => cos 27 = 10...

MathThe given expression is a product of the factors that are 2 pairs of brackets. We'll remove the brackets knowing the fact that multiplication of numbers is distributive over addition of numbers. (...

MathAccording to Cramer's Rule we can write the given set of equations in the matrix form as follows:  8 4 7   x   34   5 6 3   y  = 21   3 7 8   z  85  Now the value...

MathAccording to Cramer's Rule we can write the given set of equations in the matrix form as follows:  5 4 6  x   58  4 6 3  y  = 13   6 3 7   z   53  Now the value of...

MathGiven the inequality 10 l 10n  8 l < 80. We need to find all possible values of n. First, we will solve the same way we solve the equality. Let us divide by 10. ==> l 10n 8 l < 8 Now we...

MathGiven the expression E=1.18 *10^4 / 3*10^0 + 2*10^3/10^5. We need to rewrite into the scientific notation. Then we will rewrite into the form (a.bc)*10^d. Let us simplify the expression....