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 $k$. $M_1 = 31.5 \text{ kg}$ $L_1 = 500 \text{ m}$ $D_1 = 3 \text{ mm}$ $$ 31.5 = k \times 500 \times (3)^2 $$ $$ 31.5 = k \times 500 \times 9 $$ $$ 31.5 = 4500k $$ $$ k = \frac{31.5}{4500} = 0.007 $$ Step 3: Use the constant $k$ and the information for the second wire to find its mass $M_2$. $L_2 = 1000 \text{ m}$ $D_2 = 2 \text{ mm}$ $$ M_2 = kL_2D_2^2 $$ $$ M_2 = 0.007 \times 1000 \times (2)^2 $$ $$ M_2 = 0.007 \times 1000 \times 4 $$ $$ M_2 = 7 \times 4 $$ $$ M_2 = 28 \text{ kg} $$ The final answer is $\boxed{28 \text{ kg}}$. --- Step 1: Convert Mr. Gatua's annual salary from Ksh to K£. Given annual salary = Ksh 80,000. Since 1 K£ = 20 Ksh: $$ \text{Annual salary in K£} = \frac{80,000}{20} = 4,000 \text{ K£} $$ Step 2: Calculate the total annual tax based on the given tax rates. • First 1500 K£ taxed at 10%: $$ 1500 \times 0.10 = 150 \text{ K£} $$ • Next 1500 K£ (from 1501 to 3000 K£) taxed at 15%: $$ 1500 \times 0.15 = 225 \text{ K£} $$ • Remaining income: $4000 - 1500 - 1500 = 1000 \text{ K£}$. This falls into the 3000 - 4500 K£ bracket, taxed at 25%: $$ 1000 \times 0.25 = 250 \text{ K£} $$ Total annual tax = $150 + 225 + 250 = 625 \text{ K£}$. Step 3: Convert the total annual tax from K£ to Ksh. $$ \text{Total annual tax in Ksh} = 625 \times 20 = 12,500 \text{ Ksh} $$ Step 4: Calculate the annual personal relief. Given monthly personal relief = Ksh 1056. $$ \text{Annual personal relief} = 1056 \times 12 = 12,672 \text{ Ksh} $$ Step 5: Calculate the annual PAYE (Pay As You Earn). Annual PAYE = Total annual tax - Annual personal relief $$ \text{Annual PAYE} = 12,500 - 12,672 = -172 \text{ Ksh} $$ 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 $\boxed{\text{Ksh } 0}$. --- 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 $3 \frac{1}{3} = \frac{10}{3}$: $$ T_3 = ar^{3-1} = ar^2 = \frac{10}{3} \quad (1) $$ The sixth term ($T_6$) is $11 \frac{1}{4} = \frac{45}{4}$: $$ T_6 = ar^{6-1} = ar^5 = \frac{45}{4} \quad (2) $$ Step 3: Divide equation (2) by equation (1) to find the common ratio $r$. $$ \frac{ar^5}{ar^2} = \frac{\frac{45}{4}}{\frac{10}{3}} $$ $$ r^3 = \frac{45}{4} \times \frac{3}{10} $$ $$ r^3 = \frac{135}{40} $$ $$ r^3 = \frac{27}{8} $$ Take the cube root of both sides: $$ r = \sqrt[3]{\frac{27}{8}} $$ $$ r = \frac{3}{2} $$ Step 4: Substitute the value of $r$ into equation (1) to find the first term $a$. $$ ar^2 = \frac{10}{3} $$ $$ a\left(\frac{3}{2}\right)^2 = \frac{10}{3} $$ $$ a\left(\frac{9}{4}\right) = \frac{10}{3} $$ $$ a = \frac{10}{3} \times \frac{4}{9} $$ $$ a = \frac{40}{27} $$ The common ratio is $\frac{3}{2}$ and the first term is $\frac{40}{27}$. The final answer is $\boxed{\text{common ratio } r = \frac{3}{2}, \text{ first term } a = \frac{40}{27}}$. Got more? Send 'em 📸