A company's profit is modelled by the fraction 2x/(x-5) where x represents units sold. i. Explain what happens to the profit when x=5. Why is this significant? ii. Determine for which values of x the profit is positive.

3. c. A company's profit is modelled by the fraction (2x)/(x-5) where x represents units sold. i. Explain what happens to the profit when x=5. Why is this significant? (2 marks) When x=5, the denominator x-5 becomes 5-5=0. This makes the profit expression (2(5))/(0) = (10)/(0), which is undefined. This is significant because it indicates a critical point where the mathematical model breaks down, possibly representing a scenario where the profit cannot be determined or a point of instability in the company's operations. ii. Determine for which values of x the profit is positive. (2 marks) Step 1: For the profit to be positive, (2x)/(x-5) > 0. This occurs when both the numerator and denominator have the same sign. Case 1: Both are positive. 2x > 0 x > 0 x-5 > 0 x > 5 For both conditions to be true, x > 5. Case 2: Both are negative. 2x < 0 x < 0 x-5 < 0 x < 5 For both conditions to be true, x < 0. The profit is positive when x < 0 or x > 5. 4. a. Make the indicated variable the subject: (4 marks) i. Make y the subject of 3x + 2y = 12 Step 1: Subtract 3x from both sides. 2y = 12 - 3x Step 2: Divide both sides by 2. y = (12 - 3x)/(2) The subject is y = (12 - 3x)/(2). ii. Make r the subject of A = r^2 + 2 rh Step 1: Rearrange the equation into the standard quadratic form ar^2 + br + c = 0. r^2 + 2 hr - A = 0 Step 2: Use the quadratic formula r = -b ± sqrt(b^2 - 4ac)2a, where a=, b=2 h, and c=-A. r = -(2 h) ± sqrt((2 h)^2 - 4()(-A))2() Step 3: Simplify the expression. r = -2 h ± sqrt(4^2 h^2 + 4 A)2 r = -2 h ± sqrt(4( h^2 + A))2 r = -2 h ± 2sqrt(( h^2 + A))2 Step 4: Divide all terms by 2. r = -h ± sqrt(( h^2 + A)) The subject is r = -h ± sqrt(( h^2 + A)). 4. b. Solve the following equations: (6 marks) i. 5x - 3 = 2x + 9 Step 1: Subtract 2x from both sides. 5x - 2x - 3 = 9 3x - 3 = 9 Step 2: Add 3 to both sides. 3x = 9 + 3 3x = 12 Step 3: Divide by 3. x = (12)/(3) x = 4 The solution is x = 4. Send me the next one 📸