If f(x) = (x/2) + 3, solve for the inverse.
To solve for the inverse, the firs thing to do is change f(x) to y. y = (x/2) + 3 The next step to is get x by itself on one side of the equation. For this problem, we have to subtract 3 from both sides. y - 3 = (x/2) To fully get x by itself, we now need to multiply both sides by 2. 2(y-3) = x For simplification, the 2 needs to be distributed. x = 2y - 6 The inverse: f^(-1)(x) = 2x - 6
Glycerol (C3H8O3) is sold in drug stores as glycerine and is commonly found in soaps and shampoos. a) What is the molar mass of glycerol? b) What is the mass in grams of 1.00 mole of glycerol? c) How many molecules are in one mole of glycerol?
a) To determine the molar mass of glycerol, the chemical composition is essential. First you need to add up all the masses of the different atoms that combine to form glycerol. For each atom, you multiply the molar mass by the subscript of the number of atoms in the molecule. C = 3*(12.01g/mol) = 36.03 g/mol from C H = 8*(1.0079g/mol) = 8.0632 g/mol from H O = 3*(16.00g/mol) = 48.00 g/mol from O total molar mass = 36.03 + 8.0632 + 48.00 = 92.0932 g/mol for glycerol b) Now that you have the molar mass, it is possible to calculate the mass of 1.00 mole of glycerol using the molar mass as a conversion factor. Start with what the problem gives you and multiply by the conversion factor: 1.00 mol glycerol * (92.0932 g / 1 mol) = 92.0932 g glycerol c) To determine the number of molecules you need to use Avagadro's number: 6.022e23 For every 1 mole there are 6.022e23 molecules, therefore you have 6.022e23 molecules of glycerol in 1 mole.
Tim Allen is painting a fence at a rate of (200 – 4x) square feet per hour, where x is the number of hours since he started painting. If the fence is 800 square feet, how long will it take him to finish painting the fence? Round your answer to the nearest minute.
In order to determine how long it will take Tim to paint the fence at a rate of (200 - 4x), we need to take the interval of the rate from 0 to however many hours (h) equal to 800 square feet. To determine the hours (h), we need to start with the intergral: integral (0 --> h)= (200-4x)dx = 800 800 = [200x/1 - (4x2)/2] from 0 to h 800 = [ [200h - 2(h^2)] - [200(0) - 2(0)] ] 800 = 200h -(2h^2) Now we need to solve for h to find the amount of time it will take Tim to paint the fence. Since there are multiple h's in the equation, we need to get one side equal to zero so that we can use the quadratic formula. 0 = -2h^2 + 200h -800 Quadratic formula: x = [ (-b) +/- sqrt[(b^2) - 4ac] ] / (2a) where a is the coefficient for h^2, b is coefficient for h, and c is the remaining value (in this case, -800). After plugging the correct values in for the plus and minus squareroot: h = [ (-200) +/- sqrt[(200^2) - 4(-2)(-800)] ] / (2(-2)) h = 4.17 hours for plus value h = 95.82 hours for minus value Now it is time to decide which answers makes the most sense. Realistically, it does not make sense for it to take Tim 95 hours to paint a fence so the correct answer must be 4.17 hours. Congratulations! You have now solved a complex word problem in calculus.