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MathGiven that the area of the circle is A = 100pi cm^2. We know that the area of the circle IS : A = r^2 * pi = 100*pi ==> Divide by pi: ==> r^2 = 100 ==> r= 10 Now we will find the...

MathGiven that the average of (x , y, z and w ) = 25 Then we know that the average = sum of all numbers/ total of numbers ==> Ave = (x + y + z+w)/4 = 25 ==> multiply by 4: ==> x + y + z + w =...

MathGiven that the area of the rectangular field A is : A = 300 m^2 Also given that the perimeter P is" p = 80 m Let us assume that the sides are x and y ==> A = x*y ==> P = x+ y ==> x*y =...

MathDima buying donuts for classmates. She wants to buy 2x as many donuts as students. 18 students in...Dima wants to buy donuts for her classmates. There are 18 students in her class. She wants to buy 2x donuts for the classmates. Then she will need to buy donuts that is double the number of her...

Mathf " (x) = X^2 + X 1, f ' (0)=5, f (0)=10 First we need to determine f'(x) by calculating the integral of f''(x) We know that: f'(x) = intg f''(x) ==> f'(x) = intg (x^2 + x 1) dx =...

Mathy= tanx ( 4,1) Let the equation of the line be : y y1 = m (xx1) where (x1,y1) is any point on the line and m is the slope. ==> ( y 1) = m (x  4) Now we will calculate the slope: We know...

Mathy= x^2 y= x+2 To find the area will need to find the integral for both equations: First we need to determine the intersecting points: ==> x^2 = x+ 2 ==> x^2 x 2 = 0 ==> (x2)(x+1) = 0...

Mathf(x) = x^3  12x Find the maximum and minimum on the interval [ 3, 1] First we need to determine the critical values of f(x) in the interval [ 3, 1] Then we need to find the first derivative:...

Math2^(x3)  8 = 0 First we will add 8 to both sides: ==> 2^(x3) = 8 Now we will rewrite 8: we know that: 8 = 2^3 ==> 2^(x3) = 2^3 Now since the bases are equal, then the powers are equals...

Mathf(x) = x^2  4x 1 g(x) = 3x^2 + 5x + 4 find g(f(0) First we need to determine g(f(x) g(f(x) = g ( x^2  4x 1) = 3(x^24x1)^2 + 5(x^2  4x 1) + 4 Let us factor (x^2  4x 1) :...

MathT.V on sale for $600 . the sale price is 20 % less than regular price. What was the regular price...Given the current price of a TV = $600 The current price is the price after 20% cut: Then let us assume that x is the original price of the TV before reduction. Then : Current price = original...

Mathsin a = 45/ x cos a = 24/x We are given that sine and cosine of an angle. Then we could use the trigonometric properties to solve: We know that: sin^2 a + cos^2 a = 1 Then we will substitute:...

Math2x+ 3 <5 First we will subtract 3 from both sides: ==> 2x + 3  3 < 5  3 ==> 2x < 2 Now we will divide by 2: (remember that when dividing or multiplying by a negative number,...

Math2 1/2 inch per min. 240 inch per hour 1/4 foot per min. Which is faster? First we need to convert all of the above to the same unit. We will convert to inch per min. So we have: 2 1/2 inch/ min...

Matha) A quartic function could have two pairs of equal real zeros. It is possible .Yes it is right. Example (x2)^2 (x+i)(xi) = (xa)^2(x^2+1) has 2 ral roots. b) A cubic function could have just...

MathI think the question is to solve the equation for the value of y. Now you have given 3y + 6 = y  2 subtract y from both the sides of the equation => 3y  y + 6 = y  y 2 => 2y + 6 = 2...

MathWhat you have worked is correct. But x = 8 is not correct because , you are in the modulo 15 system. The conclusion derived from the last step 8 = 14(8)(mod15) is x = 8. The conclusion is...

MathIf a , b , c are any three integer, and if a divides bc, then implies ac divides bc. To prove or dis[rove the statement. Since abc , we can find a integer k such that bc = ak....(1) Since ac also...

MathTo prove if (ab,c) = 1 if and only (a,c)= 1 and (b,c) = 1. Since (ab,c) = 1, we can find integers m and n such that: m(ab) + nc = 1. Therefore m(ab)+nc = 1. Therefore (mb)a +nc = 1. Therefore...

MathHi, Just to share. To key x to the power of 2, usually we just key x^2 . For your question: lim (X^2  1/X) X>0 Do you mean Case 1: (x^2  1)/x or Case 2: (x^2  1/x) ? I have done...

MathThe general form of a linear function is: f(x) = ax + b If the equation can be placed into this form, then it is linear. Otherwise, it is not. 2x^2 is a quadratic (order 2) polynomial. Thus, 2x^21...

MathWe have to solve 3/(x  3)  4/(x  2) = 4 sqrt (4x) + 3 < 6. Now 3/(x  3)  4/(x  2) = 4 multiply all terms by (x3)(x2) => 3(x2)  4(x3) = 4(x3)(x2) => 3x  6  4x + 12 = 4...

MathWe have the equation [(1x)/x] +[x/ (1+x)] = 17/4 so solve for x. We notice that 17/4 can be written as 4 + 1/4 So the equation given can be written as [(1x)/x] + [x/ (1+x)] = 17/4 => [(1x)/x]...

MathI can’t give you all the ways to test for divisibility. Here are some common ones to test for divisibility by: 2: The last digit of the number is 0 or divisible by 2. 3: The sum of all the digits...

MathWhat is the length of the longest stick that can be placed in a rectangular box with dimensions 6...We are given a rectangular box with the dimensions 6 cm, 8 cm and 24 cm. We have to find the length of the longest stick that can be placed in the box. This length is equal to the diagonal of the...

MathWe need to solve x^2  4x 12 > 0. Now x^2  4x 12 > 0 => x^2 – 6x + 2x – 12 >0 => x (x – 6) + 2 (x – 6)>0 => (x – 6) (x + 2) >0 Now this is true either when both...

MathHere we need to find the length of the third side given that the length of two sides is 8 and 6 and the area of the triangle is 24 cm^2. Now the area of a triangle is given by Area =...

MathYou want to solve sqrt (x3) >= 1/ (x3). sqrt(x3) >= 1/ x3 Take the square of both the sides. => x3 >= 1/ (x3)^2 => (x3)^3 >= 1 Now (x3)^3 is greater than or equal to 1 if...

MathWhat is the value of a if the area of a circle formed by the points (7, 0), (11, 0) and (0, a) is...The area of a triangle is given by (1/2)*base*height. In the given question the vertices of the triangle are at (7, 0), (11, 0) and (0, a). The base of the triangle is equal to the distance between...

MathWe have to find solution of x^3+3x^2+4x+2 =0 Now x^3+3x^2+4x+2 =0 = x^3 + x^2 + 2x^2 + 2x + 2x + 2 =0 => x^2 ( x+1) + 2x (x+1) + 2(x+1) =0 => (x+1)(x^2 + 2x +2) =0 For x +1 = 0, x = 1. The...

MathWe notice that the value x = 1 is the solution of the equation. 2^1 + 3^1 = 5^1 2 + 3 = 5 Now, we'll have to verify if x = 1 is the only solution for the given equation or if there are more. We'll...

MathThe first one means that A is an element is B I will give you an example for each case. Let B be a set. Then B = { A, C, D ...} Then A is an element in B. The second one means that A is a subset...

MathHere we have to compare (log 11) ^2 and log 12. We assume the logarithms have the base 10. Now (log 11) ^2 can be written as (log 10*1.1) ^2 => (log 10 + log 1.1) ^2 => (1+ log 1.1) ^2 Now...

MathYour question is a little difficult to understand. I assume by origin you mean the center about which it is rotating. Now you want to know how many kilometers it has rotated at the origin. Rotation...

MathWe have to find the roots of x^6  9x^3 + 8 = 0. x^6  9x^3 + 8 = 0 => x^6  8x^3 – x^3 + 8 =0 => x^3(x^3  8)  1*(x^3  8) =0 => (x^3  1) (x^3  8) =0 Now use the difference of cubes...

MathWe have to find the roots of x^4 – 13x^2 + 36 = 0. As we have x raised to the power 4, we are going to have 4 roots. Now let us see the given equation : x^4 – 13x^2 + 36 = 0 we see that 13 can...

MathWe have to factorize x^6 – 64. Now x^6 = (x^3) ^2 and 64 = 8^2 x^6 – 64 => (x^3) ^2 – 8^2 We use x^2 – y^2 = (x  y) (x + y) => (x^3 – 8) (x^3 + 8) => (x^3 – 2^3) (x^2 + 2^3)...

MathWe have to factorize x^6 + 6x^3 + 5 First, see that x^6 + 6x^3 + 5 is of the form (x^3)^2 + 6x^3 + 5 Now if we take x^3 as y this is equal to y^2 + 6y + 5 Now 6y can be written as 5y + y => y^2...

MathTo simplify (2/x+1)  (1/x1). There are two terms each term with different denominators. The additio could be done at numerator leve if we ave common denominator. Therefore we convert both terms...

MathLet's go back to our basics. The formula for distance is given by: Distance = speed x time Rearranging, we have: Time = Distance/speed Let's work in cm 1m = 100cm So, 54m = 5400cm Substituting...

Math(No offense to Neela, but 0.35 x 60 is not 39 but 21) To convert 3.35 hours into hours and minutes, First, 60 min = 1 hr 0.35 hr = 35/100 x 1 hr = 7/20 hr = 21/60 hr = 21 min So 3.35 hr = 3 hr 21...

Mathhigher dimention problems are solved analogously with 2D extrema problems: take derivatives for each direction, and solve. Steps: Find critical points where Fx = Fy = 0; evaluate second derivative...

MathI am going to give you the equation and solve for the values invested at each interest rate. Let the amount invested by Al at the interest rate of 9% per annum . Therefore the amount invested at...

MathThe number one is not a prime. We say a positive integer p > 1 is a prime number if it has no factor other than 1 and itselef, that is p. Clearly this does not qualify 1 to be prime number as...

Mathfog(10) is the same as writing f(g(10)). Now we are given that f(x) = log x and g(x) = x^2. To find fog(10)= f(g(10)), we first find g(10). This is equal to 10^2. Now substitute g(10) = 10^2 as the...

Mathsqrt( x^2  5x + 4 ) = sqrt( x^2 + 3x 2) First we will square both sides: ==> ( x^2  5x + 4 = x^2 + 3x  2 Now let us reduce similar terms ( x^2) ==> 5x + 4 = 3x  2 Now subtract 3x from...

Mathy= 2x^2  7 y= 2x + 1 the intersection point of both functions is where both kines have the same a and y coordinates: ==> y 1= y2 ==> 2x^2  7 = 2x + 1 Now we will move all terms to the...

Mathg(x) = x^3 * cos^2 x Find g'(x) We will use the product rule to find the first derivative: Let g(x) = u * v such that: u= x^3 ==> u' = 3x^2 v= cos^2 x ==> v' = 2cos x *sinx = ...

Math2x + y = 3 ..............(1) 3y  2x = 2 We will rewrite: 2x + 3y = 2 ................(2) Now , using the elimination method we will solve the system: Let us add (1) and (2): ==> 4y = 5 Now...

MathLet the numbers be x and y : Given that the sum of the numbers is 23 Then we will wrtie: x + y = 23............(1) Also, given the difference between the numbers is 1: Then we will write: x  y =...