Line AB and line CD intersect at point E. The measure of <AEC is represented by 3x + 4 and the measure of <BED is represented by 2(2x - 6). Find the number of degrees in <AEC.
First we must draw an diagram and label it, placing the representation for the angles on the diagram. When we do that we see that these angles are vertical angles. We can tell that angles are vertical when they share a vertex, and are facing in opposite directions. Vertical angles are congruent, which means their measures are equal. We will write an equation showing that the representations for the angles are equal. 3x + 4 = 2(2x - 6) Next we will solve for x: 3x + 4 =4x - 12 (distribute the 2) 4 = x - 12 ( subtract 3x from both sides) 16 = x (add 12 to both sides) Now, using the value we found for x, substitute back into the representation for <AEC: 3(16) + 4 Simplify and find the <AEC = 52 degrees.
Using the formula A = P(1 + r)^t, find to the nearest tenth of a year how long a $100 deposit will take to double at 5% interest.
First we need to understand what all of the variables mean. "A" is the amount in the account at any given time. "P" is the principle, the initial amount deposited. "r" is the rate of interest, or growth, as a decimal and "t"is time in years. Make the appropriate substitutions: 200 = 100(1 + .05)^t Since the exponent is the variable we will be solving for, we will have to use logarithms. First we will simplify the parenthesis and divide both sides by 100: 2 = (1.05)^t Next, take the log of both sides: log2 = log 1.05^t and apply the power rule of logs, by bringing the exponent down in front to multiply the log: log2 = t(log 1.05) Lastly, solve for t by dividing both sides by log1.05: 14.20669908 = t Rounding to the nearest tenth of a year, t = 14.2 years.
Find the roots of the function y = x^2 - 5x + 6, algebraically.
The roots of a quadratic function are where the function is equal to zero. To find the roots, we will first replace the "y" with a zero. 0 = x^2 - 5x + 6 Next we will factor the right hand side: 0 = (x - 2 )(x - 3) SInce we know that the product of the two factors is equal to zero, we know that at least one of the factors is equal to zero. We will write two "baby equations" setting each factor equal to zero: 0 = x - 2 0 = x - 3 Now solve each "baby equation" one at a time. This will give the two roots of the original equation: x = 2 and x = 3