If the height that a toy airplane can fly is given by the following expression: \( -3 x^{2}-5 x+9 \). Determine the maximum height that the airplane can fly.

We are given the height function:   h(x) = –3x² – 5x + 9 Since this is a quadratic function with a negative coefficient for x² (a = –3), its graph is a downward-opening parabola. Therefore, it has a maximum value at its vertex. Step 1. Find the x-coordinate of the vertex using the formula:   x = –b / (2a) Here, a = –3 and b = –5, so:   x = –(–5) / [2(–3)] = 5 / (–6) = –5/6 Step 2. Find the maximum height by substituting x = –5/6 back into the height function:   h(–5/6) = –3(–5/6)² – 5(–5/6) + 9 First calculate (–5/6)²:   (–5/6)² = 25/36 Now substitute:   h(–5/6) = –3 × (25/36) + (25/6) + 9 Simplify each term:   –3 × (25/36) = –75/36   Convert (25/6) to a fraction with denominator 36: 25/6 = 150/36   Convert 9 = 324/36 Now, combine the terms:   h(–5/6) = (–75 + 150 + 324) / 36 = 399/36 Simplify 399/36 by dividing numerator and denominator by 3:   399 ÷ 3 = 133  and 36 ÷ 3 = 12 Thus,   h(–5/6) = 133/12 Step 3. Write the conclusion: The maximum height that the toy airplane can fly is 133/12. If you prefer a decimal approximation, 133/12 ≈ 11.08. So, the maximum height is 133/12 units.