MPM2D Quadratic Word Problems: Maximum Height & Business Revenue [Worked Solutions]

In Ontario MPM2D Grade 10 Math, quadratic word problems model projectile trajectories and business revenue using vertex form y = a(x - h)² + k. The vertex (h, k) identifies the maximum height, optimal price, or turnaround time, while the x-intercepts determine when a projectile lands or when a business breaks even.

Official Ontario Curriculum Breakdown: MPM2D

• Ministry Course Code: MPM2D (Principles of Mathematics, Grade 10, Academic)
• Prerequisite Course: MTH1W (Grade 9 Mathematics, De-streamed)
• Curriculum Strand: Quadratic Relations (Standard, Factored, and Vertex Forms)
• Core Vertex Methods: Completing the square, or finding axis of symmetry x = -b / (2a)
• Next Pathway: MCR3U (Grade 11 Functions, University Preparation)

Worked Example 1: Projectile Motion: Maximum Height of a Kicked Football

Problem: A football is kicked into the air from a tee. Its height h (in meters) above the ground after t seconds is modeled by the quadratic relation h(t) = -4.9t² + 19.6t + 0.5. Determine: (a) the maximum height reached by the football and the time it occurs, and (b) the total time the ball is in flight.

• Step 1 (Find time to maximum height using axis of symmetry): In standard form h(t) = at² + bt + c, a = -4.9, b = 19.6. Axis of symmetry t = -b / (2a) = -19.6 / (2 × -4.9) = -19.6 / -9.8 = 2.0 seconds.
• Step 2 (Calculate maximum height): Substitute t = 2.0 s into h(t): h(2.0) = -4.9(2.0)² + 19.6(2.0) + 0.5 = -4.9(4) + 39.2 + 0.5 = -19.6 + 39.2 + 0.5 = 20.1 meters.
• Step 3 (Set up equation for flight time when ball lands): Set height h(t) = 0: -4.9t² + 19.6t + 0.5 = 0.
• Step 4 (Apply quadratic formula): t = [-19.6 ± √(19.6² - 4(-4.9)(0.5))] / (2 × -4.9) = [-19.6 ± √(384.16 + 9.8)] / -9.8 = [-19.6 ± √393.96] / -9.8.
• Step 5 (Compute roots): √393.96 ≈ 19.85. t₁ = (-19.6 - 19.85) / -9.8 = -39.45 / -9.8 ≈ 4.03 seconds. t₂ = (-19.6 + 19.85) / -9.8 ≈ -0.03 seconds.
• Step 6 (State conclusion): Since time cannot be negative, discard t₂. The football reaches a maximum height of 20.1 m at 2.0 seconds, and remains in flight for approximately 4.03 seconds.

Teacher's Strategy: Completing the square yields the exact same vertex (2.0, 20.1). Factoring out -4.9 gives -4.9(t² - 4t) + 0.5 = -4.9(t - 2)² + 19.6 + 0.5 = -4.9(t - 2)² + 20.1. Show both methods on tests.

Worked Example 2: Business Revenue Optimization: Concert Ticket Pricing

Problem: A community theatre sells tickets to an Ontario arts show for $30 each, selling an average of 400 tickets. A survey shows that for every $2 increase in ticket price, 10 fewer tickets will be sold. Determine the ticket price that maximizes total revenue and find this maximum revenue.

• Step 1 (Define the independent variable x): Let x represent the number of $2 price increases.
• Step 2 (Write expressions for price and quantity): Ticket Price P(x) = 30 + 2x. Quantity Sold Q(x) = 400 - 10x.
• Step 3 (Formulate revenue equation R = Price × Quantity): R(x) = (30 + 2x)(400 - 10x) = 12000 - 300x + 800x - 20x² = -20x² + 500x + 12000.
• Step 4 (Find the vertex x-value): x = -b / (2a) = -500 / (2 × -20) = -500 / -40 = 12.5 price increases.
• Step 5 (Calculate optimal ticket price): Optimal Price = 30 + 2(12.5) = 30 + 25 = $55.00.
• Step 6 (Calculate number of tickets sold): Tickets Sold = 400 - 10(12.5) = 400 - 125 = 275 tickets.
• Step 7 (Calculate maximum revenue): Maximum Revenue = $55.00 × 275 = $15,125.00.

Teacher's Strategy: Remember that x is the number of price increases, NOT the ticket price! A very common mistake is stating $12.50 as the ticket price instead of substituting back into (30 + 2x).

The 5 Most Common Mistakes Ontario Students Make in MPM2D

• Confusing x with the final price: Stating the number of increases as the ticket price.
• Forgetting initial launch height: Assuming projectiles always start from ground level (c = 0) when h₀ is given.
• Keeping negative roots: Forgetting to discard negative time values in physical application problems.
• Sign errors when completing the square: Forgetting that factoring a negative coefficient changes the middle sign.
• Missing units on answers: Failing to include 'seconds', 'meters', or '$' on final statements.

University Admissions & Prerequisite Cutoff Impact

MPM2D is the prerequisite for MCR3U Functions. Strong mastery of quadratic applications directly enables success in Grade 11 STEM pathways.

Diagnostic Readiness Check: Test Your Knowledge

• In the relation h(t) = -5t² + 20t + 2, what represents the initial height?: Answer: 2 meters (when t = 0).
• How do you find the vertex from standard form y = ax² + bx + c?: Answer: Using the axis of symmetry x = -b / (2a), then evaluating y.
• What does the vertex represent if the parabola opens downward (a < 0)?: Answer: A maximum value.