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MathTo solve for t means to find the angle t from the given identity. We'll transform the given identity into a homogenous equation by substituting 1 by (sin t)^2 + (cos t)^2 = 1 and moving all terms...

MathFirst, we'll factorize by x the denominator: 1/(x^2+x) = 1/x(x+1) We notice that the denominator of the right side ratio is the least common denominator of 2 irreducible ratios. We'll suppose that...

MathTo extreme of a function, whose expression is a quadratic, is the vertex of the parable f(x) = y. We know that the coordinates of the parabola vertex are: V(b/2a;delta/4a), where a,b,c are the...

MathWe'll start by explaining the modulus x+4. Case 1: x+4 = x + 4 for x+4>=0 We'll solve the inequality x+4>=0: x+4>=0 x >= 4 The interval of admissible values for x is:...

MathWe'll note the point that has like coordinates as M(m,m). Since the point is located on the line y = (x  1)/2, it's coordinates verify the expression of the line. We'll put y = f(x) and we'll...

MathSince the given point is the intercepting point of the graphs of the given functions, it's coordinates verify the expressions of the functions. (1,  2) belongs to f(x)'s graph if and only if; f(1)...

MathWe'll choose the function to be integrated: f(x) = (sin x)^n We'll integrate the function: Int (sin x)^n dx We'll rewrite the given function as a product of (sin x)*(sin x)^n1. We'll write (sin...

MathThe given exponential equation requires substitution technique to compute it's roots. Now, we notice that 25 = 5^2 We'll rewrite the equation as: 5^4x  2*5^2x + 1 = 0 It is a biquadratic...

MathTo determine the antiderivative, we'll have to compute the indefinite integral of the function f(x) = (2x+5)*e^(x^2+5x) Int (2x+5)*e^(x^2+5x) dx We notice that the exponent of e is a function whose...

MathSince the two number differ by d, we assume the two numbers to be x and x+d . The sum of their reciprocals = 7, given. Therefore 1/x +1/x+3 = 7/10. Therefore multiply by 10x(x+3) both sides of...

MathThe critical number of the function is the value of x that cancels the first derivative of the function. f'(x) = (5x^2 + 7x)' f'(x) = 5*2x^1 + 7 f'(x) = 10x + 7 Now, we'll calculate the roots of...

MathI think the intended question was: How to balance the equation Al(s) + CuSO4 (aq) > Al2(SO4)3 (aq) + Cu(s) This can be done by ensuring that the total number of atoms of each element is the...

Mathlog (x^2 + 2) = log (x^2  3x + 5) We know that if log a = log b Then a = b Then we know that: x^2 + 2 = x^2  3x + 5 Reduce similar terms (x^2): ==> 2 = 3x + 5 Now solve the equation: Move...

Mathl 4x2 l < l x2 l We have 4 cases: case(1): 4x 2 < x2 ==> 3x < 0 ==> x < 0 ==> x= ( inf , 0) case(2): (4x2) < x2 4x + 2 < x2 5x < 4 ==> x < 4/5...

Mathf(x) = 3+ 3x  3x^2 To find extreme values, first we need to determine the first derivative. f'(x) = 3  6x Now we need to determine the critical values which they are the derivative's zeros....

Mathf(x) = x^2 + 5x  3 Let F(x) = integral f(x) Then the area between f , x=1 and x= 2 is: A = F(2)  F(1) Then , let us determine F(x) first. F(x) = intg f(x) = intg (x^2 + 5x 3) dx...

Mathx+ 3y = 3...............(1) 2x  y = 2..............(2) We have a system with 2 equations and 2 variable, then we will use the substitution method to solve: First let us rewrite equation no. (2)....

MathWe know that the deck of regular play card has 52 cards. These cards are: 1, 2, 3, 4, 5, 6, 7, 8,9, 10, J, Q, and K We have 4 sets of each. Now let us determine the odd numbers in the deck:...

Mathf(x) = sinx* lnx To find the first derivative, we will use the product rule: Let f(x) = u*v such that: u= sinx ==> u' = cosx v= lnx ==> v' = 1/x Now we know that: if f(x) = u*v...

Mathf(x) = x^2 + 4x  12 I assume that you need to solve for x values if f(x) = 0 ==> x^2 + 4x  12 = 0 Now we have two methods to solve: 1. Factor the equations: ==> ( x + 6) (x2) = 0 ==>...

Mathx^2 + 5x > 14 To solve the inequality, first we will have to move 14 to the left side of the inequality: ==> x^2+ 5x  14 > 0 Now we can factor the left side: ==> (x+ 7) (x2 ) >0...

MathWhat is the amount to be paid yearly to repay a loan of $100,000 given at an interest rate of 10%...To find the amount that has to be paid every year, we use the concept of present value of money. Let me explain what present value of money is. An amount A if invested for a period of n years at an...

MathIf the gcd of 414 and 33 is h, then the gcd of 414 and 33 is also h. So it is sufficient if we find the gcd h between 414 and 33 and then gcd of 414 and 33 is also h only. 414 = 33*12 + 18....

MathThe transportation cost of 1000 gallons per pipeline mile is $ 0.1. It is required to find the cost of transportation of 4million gallons for a distance of 120 miles. The procedure is to find how...

MathWe'll can also solve the system of simultaneous equations using the substitution method (instead of elimination method). For this reason, we'll extract x from the second equation: x y=1 We'll add...

Math8767 = 34*252 + 199 252 = 1*199 + 53 199 = 3*53 + 40 53 = 1*40 + 13 40 = 3*13 +1. 13 * 1*13. Therefore 1 = 40 3*13. 1 = 1*403(5340) = 4*40 3*53. 1 = 4*(1993*53)3*53 = 4*19915*53 1= 4*199...

Math1) To find the maximum or miminimum for 3x^2 in 0 < x <2. Solution: f(x) = 3x^2. f'(x) =2x f'(x) is > 0 for all x for which 0 < x< 2. So f'(x) is a, continuous increasing function....

Matha) The point estimator of the poulation mean M is the sample mean m itself.Therefore, $542.50 is the estimated value of the population mean by point estimation. b) If the sample of size n has a...

Math1) The distance travelled d+d = 2d The time taken for forward jpurney = d/20 The time taken for the return journey = d/x. So the total time taken for up and down = d/20+d/x. Therefore the average...

Math5 l 3x2 l = 10 First let us divide by 5: ==> 5 l 3x2 l /5 = 10/5 ==> l 3x 2 l = 2 Now we have two cases: case (1): 3x2 = 2 Add 2 to both sides: ==> 3x = 4 Now divide by 3: ==> x=...

MathLet the number be x and y; Given that the sum is 36 ==> x + y= 36 We will write as function of y: ==> y= 36x .............(1) Now we need to find the numbers such that their product is a...

MathWe have the point (0,2) and (2,3) passes through a line. Then we will use the slope form to determine the line. The standard for is: yya = m (xx1) such thatL (x1,y1) is any point passes through...

Mathf(x) = 3x^2 + 5x 3 To find the minimum value , first we identify the sign of x^2, since the sign if positive, that means the function has a minimum value. First we will calculte the critical...

Mathylnx  xy' = 0 First we will rewrite the equations. ylnx = xy' ==> ylnx = x *dy/dx In solving differential equation, out first priority is to isolate x an y terms on different sides. Then we...

Mathy' = sqrtx * y To solve differential equation, first we will rewrite the equation: we know that y' = dy/dx ==> dy / dx = sqrtx * y Now we will group x terms on one side and y terms on the other...

Mathf(x) = x*sinx We need to determine the indefinite integral of f(x) we note that the function is aproduct of two terms. ==> intg f(x) = intg x*sinx dx Then we will apply the rule: Let f(x) = u*dv...

Mathy= x^(e^x) To differentiate, first we will apply the natural logarithm to both sides: ==> lny = ln [x^(e^x)] We know that: ln a^b = b*ln a ==> lny = (e^x) * ln x Now we will differentiate...

Mathf(x) = cosx / (1+ sinx) To differentiate we will will assume that: u= cosx ==> we know that du = sinx v= 1+ sinx ==> v' = cosc Then, f(x) = u/v We know that , f'(x) = (u'v...

Mathy= x^3 * tanx We note that we had a product of two functions. Then we will use the product rule to solve: Let y= u*v such that: u= x^3 ==> u' = 3x^2 v= tanx ==> v' = sec^2 x Then...

MathIf f(x) is defined and continuous over the interval [a, b], except maybe at a finite number of points, we'll write Int f(x)dx from a to b as: Int f(x)dx (a>b) = Int f(x)dx (a>c) + Int...

Mathx2 = 2f(x1) + 3g(x+1).............(1) 6x + 4=4f(x1)  2g(x+1)........(2) ==> 2*(1) + (2) = ==> 2x + 4 = 4f(x1) + 6g(x+1) ==> 6x + 4 = 4f(x1)  2g(x+1) ==> 8x + 8 =...

Mathy= tan^4 x + tan^2 x First let us simplify; We will factor tan^2 x ==> y= tan^2 x ( tan^2 x + 1) ==> intg y = intg (tan^2 x ( tan^2 +1) ) dx We know that : tanx = sinx/cosx ==> intg y=...

Mathz= (3i)/(32i) First we need to rewrite the number by getting rid of the denominator. We will multiply the numerator and denominator with (3+2i) ==> z= (3i)(3+2i)/(32i)(3+2i) Let us calculate...

MathThe line passes through the point (0,2) and the slope m = 3. Let us write the equation in the standard form: yy1 = m(xx1) such that: (x1,y1) is any point that passes through the line and m is...

MathLet us calculate : The first year the sold 1200 shirts for $25 each Then , if they will sell the shirt for 27, then will sell 1150 shirts. Also, if they will sell the shirt for 29, then will...

MathWe'll write the area of a rectangular shape. A = width*length A = y*x Since only three sides of the rectangular shape must be fenced and he uses 24m of fencing, we'll write the perimeter of the...

MathThe sum of the two numbers is 72. So we assume x is a number and 72x is the other number. Their product P(x) = x(72x). We have to find the number x such that x(72x) is maximum. We know by...

MathTo convert 60 degrees to radians: We know 180 degrees = pi radians. 60 degrees = (60/180)pi radians = pi/3 radians = 1.0472 radians. To convert 156.3 degrees to radians: 180 degrees = pi radians,...

MathWe'll write the formula of the combination of n items taken k at a time: C(n,k) = n!/k!(nk)! To determine the number of sets of 6 that can be formed from 11 objects, we'll apply the combination...

MathTo find angle a , if 2cos^2a +1 = 3cosa. This is a quadratic equation in cosa. So we solve for cosa and then determine angle a. We put cosa = c in the given equation: 2c^2+1 = 3c. We subtract 3c...