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MathL(t) = 1208 ( 1.265)^t The environmentalists needs t years in order to solve the problem. Since they have enough rainbow for only 10,000 leprechauns, then they have time until the population reach...

MathGiven that: v1= 226 where D= 6 in ==> The radius (r1)= 6/2 = 3 ==> v1 = r^2 * pi * h1 ==> 226 = 9 * pi * h1...........(1) We need to find the value of the volume (v2) when the height is...

MathTo cover a certain area the builder needs x numbers of 4in tiles and y numbers of 16in tiles. If he will only use the 4in tiles he will need to replace y numbers of 16in tiles with the 4in...

MathIn the problem we are given the length of two of the sides of the triangle and we have two of the angles. The sum of the three angles of a triangle is equal to 180 degrees. Using this, the third...

MathThe general form of a circle with a center (a, b) and radius r is (x  a)^2 + (y  b)^2 = r^2 We have the center as (2 , 3) and the radius is 8 Substituting these values we get : (x + 2)^2 + ( y...

MathWe have to find the units digit of 3^4027. 3^1 = 3, 3^2 = 9, 3^3 = 27, 3^4 = 81, 3^5 = 243... We see that the units digit follows the series 3, 9, 7, 1 and it repeats after 4 powers. 3^4027 =...

MathIf the given integration is correct the derivative of (u/8)*(2u^2  a^2)sqrt( a^2  u^2) + (a^4/8)arc sin (u/a) + C should be u^2*sqrt ( a^2  u^2) [(u/8)*(2u^2  a^2)sqrt( a^2  u^2) + (a^4/8)arc...

MathWe have to find the tangent to the circle x^2 + y^2 + 10x  6y  2 = 0 parallel to y = 2x without using calculus. The given circle is x^2+y^2+10x6y2=0 => x^2+ 10x + 25 + y^2  6y + 9 = + 25 +...

MathThe line x + 8y = 9 can be rewritten in the form y = x / 8 + 9/8 where 1/8 is the slope. The angle that is formed by the line with the positive x – axis is arc tan (1/8). We need to find the...

MathThe slope of the tangent to a curve at any point is the value of the first derivative at that point. Here, we have the curve y = sqrt ( 2x^2 +1) y' = 4x * (1/2) (2x^2 + 1)^(1/2) => y' = 2x/...

MathThe area of a triangle is given by the formula (1/2)* base * height. To enclose a triangle with the largest area within a rectangle its base would have to be one of the sides of the rectangle and...

MathImagine 4inch tiles laid out on top of a 16inch square tile. There would be 4 rows of 4inch squares, with 4 tiles in each row. Each 4inch square tile has an area of 16 square inches (4 x 4),...

MathI think by x you mean one of the angles of the triangle. If we look at the relation y = sin x , the independent variable is x and the dependent variable is y. The value of sin x for an angle x is...

MathYou have the answer with you, it's the formula, you just need to substitute the numbers. I'll help you with that. Bob has invested $1000 in the bank CD, so P = $1000 The CD earns 3% interest...

MathWe have to simplify: (x^28x+16)/(x^2x12)÷(4xx^2)/(2x) Now, x^2  8x + 16 = (x  4)^2 x^2  x  12 = x^2  4x + 3x  12 => x(x  4) + 3(x  4) = (x + 3)(x  4) 4x  x^2 = x(4  x)...

MathThe replicas in Katy of the Forbidden Garden terracotta army are 1/3 scale. So a replica is 1/3 the dimensions of the original. The replica in Katy has a height of 22 inches. If the original has a...

MathWe have to solve the following set of simultaneous equations: x  2y = 6 ...(1) x + 2y = 8 ...(2) (2)  (1) => x + 2y  x + 2y = 8  6 => 4y = 2 => y = 2/4 => y = 1/2 substitute in (1)...

MathWe have to solve the following set of simultaneous equations: 2x + 4y = 1 ...(1) 4x + 9y = 2 ...(2) 2*(1)  (2) => 4x + 8y  4x  18y = 2  2 => 10y = 0 => y = 0 substitute in (2) 4x + 0...

MathWe have to solve the following set of simultaneous equations: 3x + 2y = 6 ...(1) 18x + y = 9 ...(2) 6*(1)  (2) => 18x + 12y  18x  y = 36  9 => 11y = 27 => y = 27/11 substitute in (2)...

MathWe have to prove that the functions f(x) = sqrt (x^2 + 5) and g(x) = (2*sqrt x – 1)^2 grow at the same rate as x >inf. Two functions grow at the same rate if [lim x> inf ( f(x) )] /...

MathWe have to find the derivative of f(x) = 5x * (x1)^4 We can use the product rule here: f(x) = 5x * (x1)^4 f'(x) = (5x)' * (x1)^4 + 5x * [(x1)^4]' => f'(x) = 5* ( x  1)^4 + 5x * 4*(x  1)^3...

MathWe know that the derivative of ln x = 1/x , but ln x is log(e) x. We have to find the derivative of log(10) (x^3 + 3x). First we convert it to log to the base e. y = log(10) (x^3 + 3x) => y =...

MathWe have to determine the value of Int [ (sin x)^3 dx] Int [ (sin x)^3 dx] => Int [ (sin x)^2* sin x dx] => Int [ (1 – (cos x)^2 )* sin x dx] let u = cos x => du = sin x dx => Int [...

MathWe have to find the integral of ln x. The easiest way to do this is by using integration by parts. The formula we have to substitute values in is Int [ u dv] = u*v – Int [ v du] …(1) Take u =...

MathWe have to solve (x+1)^2=2x^25x+11 for x. (x+1)^2=2x^25x+11 open the brackets => x^2 + 2x + 1 = 2x^2  5x + 11 => x^2  7x + 10 = 0 => x^2  5x  2x + 10 = 0 => x(x  5)  2(x  5) =...

MathThe model that Ariana and her father build has a scale of 0.08. Each foot in the real structure is represented as 0.08 foot in the model. The square footage of the model is 7.68 and one side is 3.2...

MathTo find the value of the given expression: 2048x^67 * 9850xy, multiply the numeric terms and the terms x and y. 2048*x^67 * 9850*x*y => 2048*9850 * x*67*x * y => 20172800 * x^(67 + 1) * y...

MathThe pilot is travelling at an airspeed of 100 km/h towards the East. The pilot has to steer the airplane to tackle the wind which is blowing. The wind is blowing at 100 km/h at an angle of 25...

MathIf the roots of the equation are equal, since the equation is a quadratic, its discriminant has to be zero. Discriminant = delta = b^2  4ac a,b,c, are the coefficients of the equation: delta =...

MathLet the total number of camels be C. We have that (1/4) of the camels had been seen in the forest. Twice the square root of the total number had gone to the forest. The number that remains is equal...

MathWe have to determine (4* sqrt 2  2 sqrt 3) (4* sqrt 2 + 2* sqrt 3) (4* sqrt 2  2 sqrt 3) (4* sqrt 2 + 2* sqrt 3) => 16* 2 + 8*sqrt 3* sqrt 2  8*sqrt 2* sqrt 3  4*3 => 32  12 => 20 The...

MathWe have to determine (2* sqrt 5 + 3*sqrt 3)^2 (2* sqrt 5 + 3*sqrt 3)^2 => (2* sqrt 5)^2 + (3*sqrt 3)^2 + 2* 2* sqrt 5*3* sqrt 3 => 4*5 + 9*3 + 12* sqrt 15 => 20 + 27 + 12*sqrt 15 => 47...

MathWe have to simplify sqrt 98 + sqrt 32  sqrt 63 sqrt 98 + sqrt 32  sqrt 63 => sqrt 49*2 + sqrt 16*2  sqrt 9*7 => 7* sqrt 2 + 4* sqrt 2  3*sqrt 7 => 11* sqrt 2  3* sqrt 7 The required...

MathGiven that log 4 (x) = 12 We need to find the values of log2 (x/4) Let us use the logarithm properties to simplify. We know that log a/b = log a  log b ==> log2 (x/4) = log2 x  log2 4 But log2...

MathThe hypotenuse of a right triangle is 2 centimeters more than the longer side of the triangle.The...Let the sides be x, y and the hypotenuse h. Given that h is 2 more than the longer sides. Let the longer side be x. ==> h = x+2...............(1) The shorter side (y) is 7 less than the longer...

MathLet the numbers be x and y. Given that the sum is 20. ==> x + y= 20 ..............(1) Also, given that the larger number is 4 less than twice the smaller. ==> x = 2y 4 ..............(2) We...

MathThe area of a rectangle is 15 square centimeters and the perimeter is 16 square centimeters. What...Given that the area of the rectangle is 15 cm^2. Also, given that the perimeter is 16 cm^2 We need to find the length of the sides. ==> Let the sides be L and w. ==> L*W = 15 ...........(1)...

MathLet the side of the second square be s, the size of the first is 5 less than the second or s  5. The area of the larger square is s^2. This is four times the area of the smaller square which is (s...

MathLet us assume that the numbers are x and y. Given that the sum of the numbers is 26. ==> x+ y= 26............(1) Also, given that the product is 165. ==> x*y = 165..........(2) We will use...

MathA company has found that when x units of a product are manufactured and sold,its revenue is given...Let the revenue be R, the profit be P, and the cost be C. Then, we know that: Profit = Revenue  cost ==> Given that R= x^2+100x Also, given that C = 240x+500 ==> P = x^2 +100x  240x 500...

MathLet f(x) = (x+3)/(2x5) We will use the quotient rule to find the derivative. We will assume that f(x) = u/v such that: u= x+3 ==> u' = 1 v = 2x5 ==> v' = 2 ==> Then, we know that...

MathThe length of a rectangular garden is 2 feet longer than 3 times its width. The perimeter of the garden is 100 feet. We have to find the width and the length of the garden. The perimeter of a...

MathThe circumference of a circle with a radius equal to r is given by the expression 2*pi*r. Here, we are given the circumference and have to find the radius. Let the radius be r 2*pi*r = 72*pi =>...

MathWe can solve y=(2+4x^2)^3 in two ways. One is use the chain rule y=(2+4x^2)^3 y' = 3*8x*(2 + 4x^2)^2 => y' = 24x*( 4 + 16x^4 + 16x^2) => y' = 96x + 384x^5 + 384x^3 Else expand (2+4x^2)^3...

MathWe have to simplify (x^2y^2+2x+1)/[(x+y)^2+2(x+y)+1] (x^2y^2+2x+1)/[(x+y)^2+2(x+y)+1] => (x^2y^2+2x+1)/(x + y + 1)^2 => (x^2 +2x+1  y^2)/(x + y + 1)^2 => [(x + 1)^2  y^2]/(x + y +...

MathThe point (2 , 3) lies on the graph f(x) and g(x). So we have 3 = a*2 + b  9 and 3 = 2*b*2  a This gives us two equations to solve for a and b. 2a + b = 12...(1) 4b  a = 3 ...(2) 4b  a = 3...

MathWe have to determine the result of the difference of fractions. This can be done by equating the denominator in both the fractions and subtracting the terms in the numerator. If possible the final...

MathWe have to find the area of the triangle with sides 8, 15, 17 without using Heron’s formula. If we have a look at the sides, see that 8^2 = 64, 15^2 = 225 and 17^2 = 289. Now, 64 + 225 = 289...

MathWe have to find lim x> inf, [1/(x  sqrt(x^2 + 2x))]. [1/(x  sqrt(x^2 + 2x))] => (x + sqrt(x^2 + 2x)/ (x  sqrt(x^2 + 2x)* (x + sqrt(x^2 + 2x) => (x + sqrt(x^2 + 2x)/ x^2  x^2  2x...

MathThe expenditure on the project is the function E(x). We have E'(x) = 12x + 3 wher x denotes the number of days since the project started. For a project that takes 6 days, the expenditure is the...