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: 1. To find the percentage error, we use the formula: Percentage Error = |Measured Value - True Value|True Value × 100\% Given: True length (L_T) = 18.6 m True width (W_T) = 12.4 m Measured length (L_M) = 18.2 m Measured width (W_M) = 12.7 m i. Measured length Step 1: Calculate the absolute error in length. Error in length = |L_M - L_T| = |18.2 - 18.6| = |-0.4| = 0.4 m Step 2: Calculate the percentage error in length. Percentage Error (Length) = (0.4)/(18.6) × 100\% ≈ 2.1505\% Rounded to two decimal places: 2.15% ii. Calculated perimeter Step 1: Calculate the true perimeter and measured perimeter. P_T = 2(L_T + W_T) = 2(18.6 + 12.4) = 2(31.0) = 62.0 m P_M = 2(L_M + W_M) = 2(18.2 + 12.7) = 2(30.9) = 61.8 m Step 2: Calculate the absolute error in perimeter. Error in perimeter = |P_M - P_T| = |61.8 - 62.0| = |-0.2| = 0.2 m Step 3: Calculate the percentage error in perimeter. Percentage Error (Perimeter) = (0.2)/(62.0) × 100\% ≈ 0.3226\% Rounded to two decimal places: 0.32% iii. Calculated area Step 1: Calculate the true area and measured area. A_T = L_T × W_T = 18.6 × 12.4 = 230.64 m^2 A_M = L_M × W_M = 18.2 × 12.7 = 231.14 m^2 Step 2: Calculate the absolute error in area. Error in area = |A_M - A_T| = |231.14 - 230.64| = 0.50 m^2 Step 3: Calculate the percentage error in area. Percentage Error (Area) = (0.50)/(230.64) × 100\% ≈ 0.2168\% Rounded to two decimal places: 0.22% 2. a) Let B be the initial number of boys and G be the initial number of girls. Step 1: Set up equations based on the initial conditions. From the first sentence, "In a class of 45 students, the number of boys is three times the number of girls." B + G = 45 (1) B = 3G (2) Step 2: Substitute equation (2) into equation (1) and solve for G. 3G + G = 45 4G = 45 G = (45)/(4) = 11.25 Step 3: Substitute the value of G back into equation (2) to find B. B = 3 × 11.25 = 33.75 Note: The number of students must be whole numbers. This problem as stated leads to non-integer values, indicating a potential flaw in the question's numerical values. However, based on the direct interpretation of the initial conditions, these are the calculated values. Initial number of boys: 33.75 Initial number of girls: 11.25 b) Solve for x in 7^2x-1 = 49^x+2. Step 1: Express both sides of the equation with the same base. 7^2x-1 = (7^2)^x+2 7^2x-1 = 7^2(x+2) Step 2: Equate the exponents. 2x-1 = 2(x+2) 2x-1 = 2x+4 Step 3: Solve for x. 2x - 2x = 4 + 1 0 = 5 This is a contradiction, which means there is no value of x that satisfies the equation. The equation has no solution. 3. a) The volume of a cylinder is 1386 cm^3. Its height is 14 cm. Find its radius. (Take = (22)/(7)) Step 1: Write down the formula for the volume of a cylinder. V = r^2 h Step 2: Substitute the given values into the formula. 1386 = (22)/(7) × r^2 × 14 Step 3: Simplify and solve for r^2. 1386 = 22 × r^2 × 2 1386 = 44 r^2 r^2 = (1386)/(44) r^2 = 31.5 Step 4: Solve for r. r = sqrt(31.5) r ≈ 5.61248... cm Rounded to two decimal places: r ≈ 5.61 cm b) The 1st, 2nd and last terms of an Arithmetic Progression (A.P.) are 5, 11 and 59 respectively. Find the number of terms in the A.P. Step 1: Identify the first term (a_1), second term (a_2), and last term (a_n). a_1 = 5 a_2 = 11 a_n = 59 Step 2: Calculate the common difference (d). d = a_2 - a_1 = 11 - 5 = 6 Step 3: Use the formula for the n-th term of an A.P., a_n = a_1 + (n-1)d, to find n. 59 = 5 + (n-1)6 Step 4: Solve for n. 59 - 5 = (n-1)6 54 = 6(n-1) (54)/(6) = n-1 9 = n-1 n = 9 + 1 n = 10 4. Table 2.1 shows the distribution of ages (in years) of 60 people in a community. To calculate the mean age and standard deviation, we first find the midpoints (x_i) for each age group and then calculate f_i x_i and f_i x_i^2. | Age (years) | Midpoint (x_i) | Frequency (f_i) | f_i x_i | x_i^2 | f_i x_i^2 | | :---------- | :--------------- | :---------------- | :-------- | :------ | :---------- | | 10 - 20 | 15 | 5 | 75 | 225 | 1125 | | 20 - 30 | 25 | 12 | 300 | 625 | 7500 | | 30 - 40 | 35 | 15 | 525 | 1225 | 18375 | | 40 - 50 | 45 | 14 | 630 | 2025 | 28350 | | 50 - 60 | 55 | 9 | 495 | 3025 | 27225 | | 60 - 70 | 65 | 5 | 325 | 4225 | 21125 | | Total | | f_i = 60 | f_i x_i = 2350 | | f_i x_i^2 = 103700 | i. Mean age Step 1: Use the formula for the mean of a grouped frequency distribution. x = ( f_i x_i)/( f_i) Step 2: Substitute the calculated sums. x = (2350)/(60) = (235)/(6) x ≈ 39.1666... years Rounded to two decimal places: x = 39.17 years ii. Standard deviation Step 1: Use the formula for the standard deviation of a grouped frequency distribution. = sqrt(( f_i x_i^2)/( f_i) - (( f_i x_i)/( f_i))^2) Step 2: Substitute the calculated sums. = sqrt((103700)/(60) - ((2350)/(60))^2) Step 3: Calculate the values inside the square root. = sqrt(1728.3333... - (39.1666...)^2) = sqrt(1728.3333... - 1533.9166...) = sqrt(194.4166...) Step 4: Calculate the square root. ≈ 13.9433... years Rounded to two decimal places: = 13.94 years 5. a) Evaluate _0^2 (3x^2 - 2x + 5) dx. Step 1: Integrate the polynomial term by term. (3x^2 - 2x + 5) dx = 3x^2+12+1 - 2x^1+11+1 + 5x + C = (3x^3)/(3) - (2x^2)/(2) + 5x + C = x^3 - x^2 + 5x + C Step 2: Evaluate the definite integral using the limits of integration. [x^3 - x^2 + 5x]_0^2 = (2^3 - 2^2 + 5(2)) - (0^3 - 0^2 + 5(0)) = (8 - 4 + 10) - (0) = 14 - 0 = 14 b) Solve for x if (x+2) + (2x-1) = 15. Step 1: Use the logarithm property A + B = (AB) to combine the terms on the left side. ((x+2)(2x-1)) = 15 Step 2: Equate the arguments of the logarithms. (x+2)(2x-1) = 15 Step 3: Expand the left side and rearrange into a quadratic equation. 2x^2 - x + 4x - 2 = 15 2x^2 + 3x - 2 = 15 2x^2 + 3x - 17 = 0 Step 4: Use the quadratic formula x = -b ± sqrt(b^2 - 4ac)2a to solve for x. x = -3 ± sqrt(3^2 - 4(2)(-17))2(2) x = -3 ± sqrt(9 + 136)4 x = -3 ± sqrt(145)4 Step 5: Check for valid solutions. The arguments of the logarithms must be positive, so x+2 > 0 x > -2 and 2x-1 > 0 x > 0.5. Thus, x must be greater than 0.5. x_1 = -3 + sqrt(145)4 ≈ (-3 + 12.04)/(4) ≈ 2.26 This solution is valid since 2.26 > 0.5. x_2 = -3 - sqrt(145)4 ≈ (-3 - 12.04)/(4) ≈ -3.76 This solution is not valid since -3.76 0.5. Therefore, the only valid solution is: x = -3 + sqrt(145)4 Handwritten Questions: sqrt(2x+6) = 10 Step 1: Square both sides of the equation. (sqrt(2x+6))^2 = 10^2 2x+6 = 100 Step 2: Solve for x. 2x = 100 - 6 2x = 94 x = (94)/(2) x = 47 sqrt(3x+1) = 4 Step 1: Square both sides of the equation. (sqrt(3x+1))^2 = 4^2 3x+1 = 16 Step 2: Solve for x. 3x = 16 - 1 3x = 15 x = (15)/(3) x = 5 What's next? Send 'em! 📸