Establish the relationship between mass (M), length (L), and diameter (D).

You're on a roll — Step 1: Establish the relationship between mass (M), length (L), and diameter (D). The mass of a wire varies jointly with its length and with the square of its diameter. This can be written as: M = kLD^2 where k is the constant of proportionality. Step 2: Use the given information for the first wire to find the constant k. M_1 = 31.5 kg L_1 = 500 m D_1 = 3 mm 31.5 = k × 500 × (3)^2 31.5 = k × 500 × 9 31.5 = 4500k k = (31.5)/(4500) = 0.007 Step 3: Use the constant k and the information for the second wire to find its mass M_2. L_2 = 1000 m D_2 = 2 mm M_2 = kL_2D_2^2 M_2 = 0.007 × 1000 × (2)^2 M_2 = 0.007 × 1000 × 4 M_2 = 7 × 4 M_2 = 28 kg The final answer is 28 kg. --- Step 1: Convert Mr. Gatua's annual salary from Ksh to K£. Given annual salary = Ksh 80,000. Since 1 K£ = 20 Ksh: Annual salary in K£ = (80,000)/(20) = 4,000 K£ Step 2: Calculate the total annual tax based on the given tax rates. • First 1500 K£ taxed at 10%: 1500 × 0.10 = 150 K£ • Next 1500 K£ (from 1501 to 3000 K£) taxed at 15%: 1500 × 0.15 = 225 K£ • Remaining income: 4000 - 1500 - 1500 = 1000 K£. This falls into the 3000 - 4500 K£ bracket, taxed at 25%: 1000 × 0.25 = 250 K£ Total annual tax = 150 + 225 + 250 = 625 K£. Step 3: Convert the total annual tax from K£ to Ksh. Total annual tax in Ksh = 625 × 20 = 12,500 Ksh Step 4: Calculate the annual personal relief. Given monthly personal relief = Ksh 1056. Annual personal relief = 1056 × 12 = 12,672 Ksh Step 5: Calculate the annual PAYE (Pay As You Earn). Annual PAYE = Total annual tax - Annual personal relief Annual PAYE = 12,500 - 12,672 = -172 Ksh Since the annual personal relief is greater than the total annual tax, Mr. Gatua pays no tax. The PAYE cannot be negative. Step 6: Calculate the monthly PAYE. Since the annual PAYE is 0, the monthly PAYE is also 0. The final answer is Ksh 0. --- Step 1: Write the general formula for the n-th term of a geometric series. T_n = ar^n-1 where a is the first term and r is the common ratio. Step 2: Use the given information to form two equations. The third term (T_3) is 3 (1)/(3) = (10)/(3): T_3 = ar^3-1 = ar^2 = (10)/(3) (1) The sixth term (T_6) is 11 (1)/(4) = (45)/(4): T_6 = ar^6-1 = ar^5 = (45)/(4) (2) Step 3: Divide equation (2) by equation (1) to find the common ratio r. (ar^5)/(ar^2) = (45)/(4)(10)/(3) r^3 = (45)/(4) × (3)/(10) r^3 = (135)/(40) r^3 = (27)/(8) Take the cube root of both sides: r = 3/(8) r = (3)/(2) Step 4: Substitute the value of r into equation (1) to find the first term a. ar^2 = (10)/(3) a((3)/(2))^2 = (10)/(3) a((9)/(4)) = (10)/(3) a = (10)/(3) × (4)/(9) a = (40)/(27) The common ratio is (3)/(2) and the first term is (40)/(27). The final answer is common ratio r = (3)/(2), first term a = (40)/(27). Got more? Send 'em 📸