(b) The base of a right angled triangle is 5 cm less than 3 times the height. The area of the triangle is \( 14 \mathrm{~cm}^{2} \). What is the height of the triangle? (c) Andrew buys a number of pens for a total of R300. He finds that 3 of the pens are defective and sells all the remaining pens for R10 more than he originally paid. He still manages to make an overall profit of R15. What was the original price he bought the pens for?

We will solve each part step by step. ───────────────────────────── Part (b) The problem states that in a right‐angled triangle the base is 5 cm less than 3 times the height. If we let the height be h, then the base is (3h – 5). The area of a right‐angled triangle is given by   Area = ½ × base × height Setting up the equation using the given area 14 cm²:   ½ × h × (3h – 5) = 14 Multiply both sides by 2 to eliminate the fraction:   h(3h – 5) = 28 Expanding the left-hand side:   3h² – 5h = 28 Rearrange the equation to standard quadratic form:   3h² – 5h – 28 = 0 To solve the quadratic, compute the discriminant (Δ):   Δ = (–5)² – 4 × 3 × (–28) = 25 + 336 = 361 Since 361 is a perfect square (19²), apply the quadratic formula:   h = (5 ± 19) / (2 × 3) Taking the positive solution (since height must be positive):   h = (5 + 19) / 6 = 24 / 6 = 4 cm Thus, the height of the triangle is 4 cm. ───────────────────────────── Part (c) Andrew buys a number of pens for a total of R300. Let n be the number of pens he bought. His cost per pen is:   Cost per pen = 300 / n However, 3 of the pens are defective, so he sells only (n – 3) pens. He sells each at R10 more than what he originally paid per pen. Therefore, the selling price per pen is:   Selling price per pen = (300/n) + 10 The total revenue from selling the (n – 3) pens is:   Revenue = (n – 3) × [(300/n) + 10] We’re told that after selling the pens he makes an overall profit of R15. Since his total cost was R300, his total revenue must be:   Total revenue = 300 + 15 = R315 So, we have the equation:   (n – 3)[(300/n) + 10] = 315 To solve, first eliminate the fraction by multiplying through by n:   (n – 3)(300 + 10n) = 315n Expand the left-hand side:   300(n – 3) + 10n(n – 3) = 315n   => 300n – 900 + 10n² – 30n = 315n Combine like terms:   10n² + (300n – 30n) – 900 = 315n   => 10n² + 270n – 900 = 315n Subtract 315n from both sides to set the equation to zero:   10n² + 270n – 315n – 900 = 0   => 10n² – 45n – 900 = 0 Divide the entire equation by 5 to simplify:   2n² – 9n – 180 = 0 Now, solve the quadratic using the quadratic formula, where a = 2, b = –9, and c = –180:   n = [9 ± √( (–9)² – 4×2×(–180) )] / (2×2)   => n = [9 ± √(81 + 1440)] / 4   => n = [9 ± √1521] / 4 Since √1521 = 39, we have:   n = (9 ± 39) / 4 We discard the negative solution as n must be positive:   n = (9 + 39) / 4 = 48/4 = 12 Thus, Andrew originally bought 12 pens. The original price per pen was:   300 / 12 = R25 ───────────────────────────── Final Answers (b) The height of the triangle is 4 cm. (c) The original price Andrew paid per pen was R25.