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Math358x = 17(mod313). To find the solution x, where x is in Z. We proceed like this: Let us find the gcd in the usual procedure: 358 = 313*1 +45. 313 = 45*6+43 45 = 43*1+2. 1= 2*21 +1. Therefore , we...

MathHow many TVs will give maximum profit and what is the maximum profit?The daily profit, P, made by...The function for the profit earned per TV manufactured is given as P = 100x − 0.02x^2. Now we need to find the number of TV's that should be manufactured to earn the maximum profits. To solve...

MathLet the length of the rectangle be L and the width be W Given the the Length is twice the width: Then, we will write: L = 2W.............(1) Also, given that the area is 98. We know that the area...

MathLet the equation of the line be : y y1 = m ( xx1) where ( x1, y1) any point passes through the line and m is the slope. Given that the line passes through the origin point Then the line passes...

MathLet mark's age be M and his sister's age is S Given that mark is 4 years older than his sister: Then M = S + 4 ............(1) Now after 3 years: Mark's age = M + 3 His sister's age = S + 3 But...

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

MathLet the vertices of the triangle be : A(1,1) B( 2,0) and C ( 4,2) First we will calculate the length of the sides: AB = sqrt( 2+1)^2+ ( 01)^2 = sqrt(3^2 + 1) = sqrt( 10) ==> AB...

Math2 < 2x4 < 6 To solve the inequality, first we need to solve the same ways we solve the equality. We need to isolate x by itself : Let us add 4 to all sides: ==> 2 + 4 < 2x4 + 4 <...

Math3z  2i = (5+i) / ( 1i) First we need to rewrite z using the form z =a+ bi First we will multiply and divide the right side by ( 1+ i) ==> 3z2i = ( 5+ i)(1+i) / (1i)(1+i) ==> 3z  2i = (...

MathGiven the points A(a,3) B (4, a+3) Also, given that the distance between A and B is 4 units. We will use the distance formula to solve: We know that: D = sqrt[ ( xBxA)^2 + ( yByA)^2 ] ==>...

MathGiven that the volume of the cylinder = 38 cm^3 Also given the height of the cylinder = 3 cm We need to calculate the circumference of the base : First we will calculate the radius by using the...

Mathf(x) = 3x^3 + 5x^2  3x + 6 First, we will find the integral of f(x). Let F(x) = integral f(x) Then, the definite integral is given by: F = F(1)  F(0) Let us calculate. F(x) = intg f(x) =...

MathYes

MathWe know that the volume of the cylinder is: V = r^2 * pi * h where r is the radius and h is the height. Given that the height is h = 7 ==> V = r^2 * pi * 7 Now to determine the radius ( r), we...

MathGiven the derivative, f;(x) = 3x^2  6x + 3 Also, given f(0) = 4. We need to determine the function f(x). We know that: f(x) = integral f'(x) : Then, let us integrate f'(x)" ==> f(x) = intg...

MathAB = 2x3 and AC = 2x+4 and BC = 7 find x if AC is the hypotenuse. I am assuming that ABC is a right angle triangle where AC is the hypotenuse. Then we know that: AC^2 = ( AB^2 + BC^2 Let us...

MathWe are given that ( 3,4) is the midpoint of AB and the coordinates of B are (1,6). We have to find the coordinates of A. Let the coordinates of A be (x, y) So as (3 , 4) is the mid point of A and...

MathGiven the functions: f(x) = log x^2. g(x) = x+ 5. We need to find f(g(5)). First, we need to determine f(g(x)), then substitute with x= 5 f(g(x)) = f( x+ 5) Now we will substitute with x+ 5 in...

Math2x+ y = 11.............(1) x^2  y^2 = 24............(2) First we will rewrite (1): 2x + y = 11 ==> y= 11 2x Now we will substitute in (2): ==> x^2  y^2 = 24 ==> x^2  ( 112x)^2 = 24...

MathA box with a square base is made using 1000 ft^2 of material. Now we have to find the maximum possible volume of the box. We know that the volume of the box would be V=x^2*y, where x is the side of...

MathThe length of the journey is 2350 km. The time taken by the airplane is 3 hrs and 15 minutes. Now 3 hours 15 minutes is equal to 3.25 hrs. Therefore the speed of the plane is 2350 / 3.25 = 723.07...

MathMaria has $350, she spends 2/7 of her money & then invests the rest for 2 years at 5% per year simple interest. We need to find how much money does Maria have at the end of the 2 years? As...

MathLet the length of the rectangle be L and the width be W: Given that the length is 3 more than twice the width: ==> we will write" L= 3 + 2W..............(1) Also, given the perimeter is : P = 18...

Maths=t^36t^2+qt+1. To analyse the motion. When t = 0, the initial distance covered is 1. ds/dt = 3t^2 6t +q., When t = 0, the initial velocity = q. d2x/dt^2 = 6t6. So the acceleration is zero when...

MathThe width of a rectangle is 45% of the length. If the perimeter of the rectangle is 58cm, what is...We'll consider x as the length of the rectangle. We'll note the width of the rectangle as w. We know, from enunciation that the width is: w = 45*x/100 Now, we'll write the formula of the perimeter...

MathFrom enunciation, we conclude that we have two points (2,9) and (3,9) that are located on the graph of the linear function f(x). We'll write the linear function in the point slope form: f(x) = mx...

MathJohn is writing an article. The article is limited to 48000 words. He decides to use two types of...Large font ==> 1800 words / page Small font ==> 2400 words/ page Total pages = 21 and 48000 words Now let us assume that: x = number of pages with Large Font y= number of pages with small...

MathLet the function be : f(x) = ax^2 + bx + C Given the points (1,4), (1, 2) and (2,1) are on the curve of f(x): Then the points should verify the equation such that: f(1) = 4 f(1) = 2 and...

MathThere are 3 ways, at least, to find the vertex of a parabola. We'll write the function as: f(x) = a(xh)^2 + k, where the vertex has the coordinates v(h,k) We'll write the given function: f(x) =...

MathThe area of a square is 45 more square inches than its perimeter. What is the length of each side...Let x be one side of the square. We'll write the formula for the area of the square: A = x^2 We'll write the formula for the perimeter of the square: P = 4x Now, we'll write mathematically the...

MathWe'll rewrite the inequality: log 3 5 + 1/log 3 5 > 2 We'll note log 3 5 =a We'll rewrite the inequality in a: a + 1/a > 2 We'll multiply by 2 the inequality: a^2 + 1 > 2a We'll subtract...

MathThe equation of the slant asymptote is put in the standard form and it is written as: y = mx + n, where m is the slope of the asymptote and n is the y intercept. The equation of the slant asymptote...

MathYour question is already in slopeintercept form. slopeintercept looks like this: y = mx + b which looks like yours with m = 1/2 and b = 1. If instead you are supposed to transform the equation...

MathOne way to solve the inequality is to draw a graph of the function f(x) = 3x^29x+5 The area between the roots is the area where the graph goes below the x axis. This area represents the solution...

MathWe'll impose the constraints of existence of logarithms: 1) x+2 > 0 2) x + 2 different from 1. 3) 15 + 2x x^2 > 0 We'll solve the first constraint: 1) x+2 > 0 We'll subtract 2 both...

MathThe sides of a triangle measure 9, 15, and 18.If the shortest side of a similar triangle measures...The ratio of sides of the first triangle is 9:15:18. The ratio of sides of the similar triangle is 6:X:Y. Now, we'll determine the proportion: 6/9 = a We'll cross multiply and we'll get: 9a = 6...

MathFirst,we'll calculate the perimetero fthe original triangle, adding up the three lengths of the sides: P1 = 8+9+10 P1 = 27 Now, we'll divide the perimeter of the similar triangle, namely 81, by the...

MathFor f(x) to be increasing, the derivative of f(x) has to be positive. We'll determine the first derivative: f'(x) = (ax^3+bx^2+cx+d)' f'(x) = 3ax^2 + 2bx + c For the expression of the first...

MathWe notice that if we'll add log 12 (3)=a and log 12 (5)=b, we'll get: log 12 (3) + log 12 (5) = a + b (1) Since the bases are matching, we'll apply the rule of product: log 12 (3) + log 12 (5) =...

MathTo prove that A is a constant means that to prove that the result of the ratio does not depend on x. We notice that the numerator is a sum of logarithms that have matching bases. We'll use the rule...

MathWe are given the function f(x) = (3x  12) / (x  4). A function is continuous at x= c if: f(c) exists. lim x>c f(x) exists. lim x>c f(x) = f(c). Now we see that for the function f(x) =...

MathHere the amount of air in the balloon at any moment of time t is given as V(t). Now if the balloon is being filled or not can be determined by finding the value of the differential of the function...

MathWe need to factor the following equations: 1. 2x^2+3x2=0 ==> ( 2x 1) ( x+ 2) = 0 ==> x1= 1/2 ==> x2= 2 2. 2x^23x9=0 ==> ( 2x +3) ( x 3) = 0 ==> x1= 3/2 ==> x2= 3 3....

MathGiven the functions: f(x) = 12  4x g(x) = 1/x We need to find the domain of g(f(x)). First we need to determine g(f(x)) g(f(x)) = g( 12  4x). Now we will substitute ( 12  4x ) in g(x). ==>...

MathGiven the function f(x) = 3x  7. We need to find the inverse function f^1 (x). First we will rewrite. Let y = 3x  7. The goal is to isolate x on one side. Now we will add 7 to both sides. ==>...

MathGiven the functions f(x) and h(x) such that: f(x) = x^2 + x h(x) = x / (1x). We need to find h(f(x)) To find h(f(x) we will substitute with f(x) in g(x). h(f(x)) = h ( x^2 + x) Now substitute with...

MathGiven that the number of boys and girls in a class is as follows: Number of boys = 11 Number of girls = 9 Then we conclude that the total number of students in the class= 20 students. Then, the...

MathGiven the line segments AB, BC and AC such that: AB = 4x  15 BC = 2x + 3 AC = 48. Also given that AC is a hypotenuse. Then BC and AB are the other sides of the right angle triangle. Then we know...

MathWhen adding a zero to the numbers, the mean will get smaller. Let us prove it by an example. For example: Let "2, 3, 5, 7, 9" be numbers that their mean is as follow. m = (2+ 3 + 5+ 7+ 9) / total...

MathGiven the equation 2x  3y = 30. We need to find the parallel slope to the given line. First we need to rewrite the equation of the line into the slope format. y= mx + b ==> 2x  3y = 30 ==>...