Determine the equation of the tangent to the circle with equation (x-3)2 + (y-5)2 = 5 at the point (6, 7).

Here are the solutions to questions 10, 11, and 12. 10. Determine the equation of the tangent to the circle with equation (x-3)^2 + (y-5)^2 = 5 at the point (6, 7). Step 1: Identify the center of the circle. The equation of the circle is (x-3)^2 + (y-5)^2 = 5. The center of the circle is (h, k) = (3, 5). Step 2: Calculate the gradient of the radius connecting the center to the point of tangency. The point of tangency is (x_1, y_1) = (6, 7). The center is (x_c, y_c) = (3, 5). The gradient of the radius m_r is: m_r = (y_1 - y_c)/(x_1 - x_c) = (7 - 5)/(6 - 3) = (2)/(3) Step 3: Determine the gradient of the tangent line. The tangent line is perpendicular to the radius at the point of tangency. If m_t is the gradient of the tangent, then m_t × m_r = -1. m_t = -(1)/(m_r) = -(1)/(2/3) = -(3)/(2) Step 4: Find the equation of the tangent line using the point-slope form. Using the point (6, 7) and the gradient m_t = -(3)/(2): y - y_1 = m_t (x - x_1) y - 7 = -(3)/(2) (x - 6) Multiply both sides by 2 to eliminate the fraction: 2(y - 7) = -3(x - 6) 2y - 14 = -3x + 18 Rearrange the equation into the form Ax + By + C = 0: 3x + 2y - 14 - 18 = 0 3x + 2y - 32 = 0 The equation of the tangent is: 3x + 2y - 32 = 0 11. The volume of a cylinder is given by V = r^2 h. Find the percentage change in V if r increases by 8% and h decreases by 12%. Step 1: Express the new radius and height in terms of the original radius and height. Let the original radius be r_0 and the original height be h_0. Original volume V_0 = r_0^2 h_0. Radius increases by 8%: New radius r_1 = r_0 + 0.08 r_0 = 1.08 r_0. Height decreases by 12%: New height h_1 = h_0 - 0.12 h_0 = 0.88 h_0. Step 2: Calculate the new volume V_1. V_1 = r_1^2 h_1 Substitute the expressions for r_1 and h_1: V_1 = (1.08 r_0)^2 (0.88 h_0) V_1 = (1.08^2) r_0^2 (0.88) h_0 V_1 = (1.1664 × 0.88) r_0^2 h_0 V_1 = 1.026432 r_0^2 h_0 Since V_0 = r_0^2 h_0, we have: V_1 = 1.026432 V_0 Step 3: Calculate the percentage change in volume. Percentage change = (V_1 - V_0)/(V_0) × 100\% Percentage change = (1.026432 V_0 - V_0)/(V_0) × 100\% Percentage change = (1.026432 - 1) × 100\% Percentage change = 0.026432 × 100\% Percentage change = 2.6432\% The percentage change in volume is: 2.64\% increase (rounded to two decimal places). 12. A trader deals in two types of rice; rice A and rice B. Type A rice costs Kshs. 400 per bag and type B costs Kshs. 350 per bag. The trader mixes 30 bags of type A with 50 bags of type B. If he sells the mixture at a profit of 20%, calculate the selling price of one bag of the mixture. Step 1: Calculate the total cost of rice A. Cost of 30 bags of type A = 30 bags × Kshs. 400/bag = Kshs. 12000. Step 2: Calculate the total cost of rice B. Cost of 50 bags of type B = 50 bags × Kshs. 350/bag = Kshs. 17500. Step 3: Calculate the total cost of the mixture. Total cost of mixture = Cost of type A + Cost of type B Total cost of mixture = Kshs. 12000 + Kshs. 17500 = Kshs. 29500. Step 4: Calculate the total number of bags in the mixture. Total bags = 30 bags (Type A) + 50 bags (Type B) = 80 bags. Step 5: Calculate the cost price per bag of the mixture. Cost price per bag = Total cost of mixtureTotal bags = Kshs. 2950080 = Kshs. 368.75. Step 6: Calculate the selling price per bag with a 20% profit. Selling price per bag = Cost price per bag × (1 + Profit percentage) Selling price per bag = Kshs. 368.75 × (1 + 0.20) Selling price per bag = Kshs. 368.75 × 1.20 Selling price per bag = Kshs. 442.50. The selling price of one bag of the mixture is: Kshs. 442.50