This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
b) Step 1: Determine the total number of balls and the number of each color. Number of red balls = 3 Number of black balls = 4 Total number of balls = 3 + 4 = 7 Step 2: Identify the possible ways to select two different balls. There are two scenarios for selecting two different balls: 1. First ball is Red, second ball is Black (RB). 2. First ball is Black, second ball is Red (BR). Step 3: Calculate the probability for the first scenario (RB). The probability of selecting a red ball first is (3)/(7). Since the first ball is not replaced, there are now 6 balls left (2 red, 4 black). The probability of selecting a black ball second is (4)/(6). P(RB) = P(R first) × P(B second | R first) = (3)/(7) × (4)/(6) = (12)/(42) Step 4: Calculate the probability for the second scenario (BR). The probability of selecting a black ball first is (4)/(7). Since the first ball is not replaced, there are now 6 balls left (3 red, 3 black). The probability of selecting a red ball second is (3)/(6). P(BR) = P(B first) × P(R second | B first) = (4)/(7) × (3)/(6) = (12)/(42) Step 5: Add the probabilities of the two scenarios to find the total probability. The probability of selecting two different balls is the sum of the probabilities of these two mutually exclusive events. P(different balls) = P(RB) + P(BR) = (12)/(42) + (12)/(42) = (24)/(42) Step 6: Simplify the fraction. (24)/(42) = (4 × 6)/(7 × 6) = (4)/(7) The probability of selecting two different balls is (4)/(7). a) Step 1: Convert the mixed fraction to an improper fraction. 2(1)/(4) = (2 × 4 + 1)/(4) = (8+1)/(4) = (9)/(4) Step 2: Rewrite the given logarithmic equation. _x (9)/(4) = 2 Step 3: Convert the logarithmic equation to an exponential equation. By the definition of logarithms, if _b a = c, then b^c = a. Applying this definition: x^2 = (9)/(4) Step 4: Solve for x. Take the square root of both sides: x = ±sqrt((9)/(4)) x = ±sqrt(9)sqrt(4) x = ±(3)/(2) Step 5: Consider the domain of the logarithm. For _x A to be defined, the base x must be positive and x ≠ 1. Therefore, x = -(3)/(2) is not a valid solution. The only valid solution is x = (3)/(2). The value of x is (3)/(2).