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.
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here are the solutions to the problems: 13) To find the slope of the line segment joining points A(-2, -4) and B(3, 1), use the slope formula m = (y_2 - y_1)/(x_2 - x_1). Step 1: Identify the coordinates. (x_1, y_1) = (-2, -4) (x_2, y_2) = (3, 1) Step 2: Substitute the values into the formula. m = (1 - (-4))/(3 - (-2)) m = (1 + 4)/(3 + 2) m = (5)/(5) m = 1 The correct option is B. 14) Two sets are equal if they contain exactly the same elements. Step 1: List the elements of each set. X = \a, b, 1, 2, 3\ Y = \2, d, b, c\ Z = \1, b, c, d\ W = \b, 2, 1, 3, a\ Step 2: Compare the sets. • Option a) Y ≠ Z because 2 Y but 2 Z (and 1 Z but 1 Y). • Option b) X ≠ Z because a X but a Z (and c, d Z but c, d X). • Option c) Y ≠ W because d, c Y but d, c W (and 1, 3, a W but 1, 3, a Y). • Option d) X = W because both sets contain the elements \a, b, 1, 2, 3\. The order of elements does not matter for set equality. The correct option is D. 15) Given A = (3)/(4). In a right-angled triangle, A = oppositeadjacent. Step 1: Identify the opposite and adjacent sides. Opposite = 3 Adjacent = 4 Step 2: Use the Pythagorean theorem to find the hypotenuse. Hypotenuse^2 = Opposite^2 + Adjacent^2 Hypotenuse^2 = 3^2 + 4^2 Hypotenuse^2 = 9 + 16 Hypotenuse^2 = 25 Hypotenuse = sqrt(25) = 5 Step 3: Find A. In a right-angled triangle, A = oppositehypotenuse. A = (3)/(5) The correct option is D. 16) To find the mean (average) score, sum all the scores and divide by the number of scores. Step 1: Sum the scores. Sum = 78 + 81 + 69 + 82 + 75 = 385 Step 2: Divide by the number of scores, which is 5. Mean = (385)/(5) = 77 The correct option is B. 17) To solve the proportion (r+1):7 = 20:28: Step 1: Write the proportion as an equation of fractions. (r+1)/(7) = (20)/(28) Step 2: Simplify the fraction on the right side. (20)/(28) = (5 × 4)/(7 × 4) = (5)/(7) Step 3: Set the simplified fractions equal and solve for r. (r+1)/(7) = (5)/(7) Since the denominators are equal, the numerators must be equal. r+1 = 5 r = 5 - 1 r = 4 The correct option is A. 18) To find the volume of a right cylinder, use the formula V = r^2 h. Step 1: Identify the given values. Base diameter = 28 cm, so radius r = (28)/(2) = 14 cm. Height h = 40 cm. Use = (22)/(7). Step 2: Substitute the values into the formula. V = (22)/(7) × (14 cm)^2 × 40 cm V = (22)/(7) × 196 cm^2 × 40 cm Step 3: Perform the calculation. V = 22 × (196)/(7) × 40 cm^3 V = 22 × 28 × 40 cm^3 V = 616 × 40 cm^3 V = 24640 cm^3 The correct option is C. 19) This question requires data from a table or context not provided in the image. Therefore, it cannot be answered. 20) This question requires data from a table or context not provided in the image. Therefore, it cannot be answered. 21) To find the sum of 8144_9 and 1542_9: Step 1: Add the numbers column by column from right to left, remembering that this is base 9 (digits 0-8). r 8144_9 \\ + 1542_9 \\ Step 2: Add the rightmost column (4+2). 4 + 2 = 6_9 Step 3: Add the next column (4+4). 4 + 4 = 8_9 Step 4: Add the next column (1+5). 1 + 5 = 6_9 Step 5: Add the leftmost column (8+1). 8 + 1 = 9 Since 9 is equal to the base, write 0 and carry 1. 9 = 1 × 9 + 0 10_9 So, the sum is 10686_9. The correct option is A. 22) To find the prime factorization of 588: Step 1: Divide by the smallest prime number, 2. 588 ÷ 2 = 294 Step 2: Divide by 2 again. 294 ÷ 2 = 147 Step 3: Divide by the next smallest prime number, 3. 147 ÷ 3 = 49 Step 4: Divide by the next smallest prime number, 7. 49 ÷ 7 = 7 Step 5: The prime factors are 2, 2, 3, 7, 7. 588 = 2^2 × 3^1 × 7^2 The correct option is D. 22) (This is a duplicate question number, but a new problem) To find Mr. Grabrah's commission: Step 1: Identify the selling price and commission rate. Selling price = \43,770.00 Commission rate = 3\% Step 2: Calculate the commission. Commission = 3\% of \43,770.00 Commission = 0.03 × \43,770.00 Commission = \1313.10 The correct option is C. 23) To calculate the simple interest, use the formula I = PRT. Step 1: Identify the principal (P), rate (R), and time (T). P = \37.00 R = 2\% = 0.02 T = 9 years Step 2: Substitute the values into the formula. I = \37.00 × 0.02 × 9 Step 3: Perform the calculation. I = \0.74 × 9 I = \6.66 The correct option is C. 24) The logarithm property for the product of two numbers is _b (XY) = _b X + _b Y. Step 1: Apply the product rule for logarithms. _b MN = _b M + _b N The correct option is A. 25) To evaluate the expression P(1+rt) with the given values: Step 1: Substitute P=2, r=-5, and t=3 into the expression. P(1+rt) = 2(1 + (-5)(3)) Step 2: Perform the multiplication inside the parenthesis first. 2(1 + (-15)) Step 3: Perform the addition inside the parenthesis. 2(1 - 15) 2(-14) Step 4: Perform the final multiplication. 2 × (-14) = -28 The final answer is -28. Got more? Send 'em!