Solution library

The water lily takes 47 days to cover half the pond, given that it doubles in size each day and covers the entire pond on day 48.

The equation 3^(x^2)/9^x = 81 is solved by rewriting the equation with the same base, simplifying the denominator, applying the quotient rule for exponents, equating the exponents, rearranging into a quadratic equation, and finally using the quadratic formula to find the two solutions, x = 1 + √5 and x = 1 - √5.

The equation x^2 + 3x + 7 = 6/(x^2 + 3x + 2) is solved by substituting u = x^2 + 3x, resulting in a quadratic equation in u. Factoring the quadratic equation yields two possible values for u, which are then substituted back into the substitution to find the solutions for x using the quadratic formula. One case results in two real solutions, while the other yields no real solutions.

The solution to the equation \sqrt{7 + \frac{3}{\sqrt{x}}} = 7 - \frac{9}{x} is x = \frac{9}{4}.

The solution demonstrates how to find the value of RS by expanding the second equation, substituting the first equation, and simplifying the resulting equation.

The solution for X in the equation 2^X + 2 = 66 is 6.

To solve for a in the equation (a/2) ÷ (5/2) = 2, first rewrite the division as multiplication by the reciprocal, then multiply the numerators and denominators, simplify the fraction, isolate a by multiplying both sides by 10, and finally solve for a by dividing both sides by 2, resulting in a = 10.

To solve the equation 16/x = x^2/4, cross-multiply to get 64 = x^3, then take the cube root of both sides to find x = 4.

The solution to the equation (1/3)^(x+3) = 9^x is x = -1.

The inverse Laplace transform of 1/x^(3/2) is 2/(sqrt(pi))*sqrt(x).

The Laplace transform of sin^2(t) is 2 / (s * (s^2 + 4)).

The Laplace transform of e^(t/2) is 1/(s - 1/2), provided s > 1/2.

The equation |a - 5| = |5 - a| is proven by considering two cases: when (a - 5) is greater than or equal to zero and when (a - 5) is less than zero. In both cases, the equation holds true, demonstrating that the absolute value of the difference between a and 5 is equal to the absolute value of the difference between 5 and a for all values of a.

The rational function 1/(3x^2 + 4x + 1) is decomposed into partial fractions as (3/2)/(3x + 1) - (1/2)/(x + 1) by factoring the denominator and solving for the unknown constants using strategic values of x.

The partial fraction decomposition of the rational expression x/((x+1)(x-4)) is (1/5)/(x+1) + (4/5)/(x-4).

The area of the region defined by the inequalities y ≥ x, y > -x + 1, and y < 10 is 90.25 square units.

The system of equations is solved by substitution, expanding, simplifying, using the quadratic formula to find x values, and then substituting back into the equations to find the corresponding y values. The solutions are (3, 4) and (-4, -3).

The solution to the system of equations x + 2y = 5 and 3x - y = 1 is x = 1 and y = 2, found using the elimination method.

The solution for x in the equation involving fractions with (x-1) and (x+1) in the denominator is x = √2 and x = -√2.

The solution to the equation 3^(a^2 - 10a + 25)! = 729^4 is a = 3 or a = 7, found by equating exponents after expressing both sides with the same base and solving the resulting quadratic equation.

To find the difference between ba and ab, where a and b are digits, we analyze the units and hundreds digits, leading to the conclusion that the hundreds digit of the difference is 3.

The solution to the equation 4a³ = 2⁵ is a = 2.

The equation was solved by simplifying the square root, isolating the square root term, squaring both sides, moving all terms to one side, factoring out a common factor, and solving for x, resulting in two solutions: x = 0 and x = 16/81.

The solution to the equation 3^x = 1 and 3^y = 81 is x + y = 4.

By solving the system of equations x + y = 3 and x - y = 9, we find x = 6 and y = -3. Then, substituting these values into (x/y)^2, we get 4.

The total volume of infinitely many cubes with side lengths decreasing according to the sequence 1, 1/√2, 1/√3, ... is finite, while the total surface area is infinite.

The expression 2^(n+4) - 2(2^n) / 2(2^(n+3)) simplifies to 7/8.

The derivative of xy + sin(x) = e^y with respect to x is found using implicit differentiation, applying the product rule and chain rule, and solving for dy/dx.

The derivative of x² + y² = 25 with respect to x is -x/y.

The derivative of sin(2x² + 3x) is found using the chain rule, differentiating the outer sine function and the inner quadratic function separately, and then multiplying the results: cos(2x² + 3x) * (4x + 3).