Question: A public health communicator is analyzing maternal health data from three clinics in Vancouver. If the number of prenatal visits at each clinic is modeled by the function $ f(x) = ax^3 + bx^2 + cx + d $, and the data shows $ f(1) = 15, f(2) = 30, f(3) = 45, f(4) = 60 $, find $ f(5) $.

["Title: Predicting Maternal Health Outcomes: A Cubic Model for Prenatal Visits in Vancouver Clinics", "Meta Description:\nA public health communicator uses a cubic function to model prenatal visit data from three Vancouver clinics. With known values at four points, we determine the number of visits at the fifth clinic point using predictive modeling.", "Introduction\nUnderstanding maternal health outcomes is critical to improving prenatal care. In a recent analysis, a public health communicator in Vancouver modeled prenatal visit data using a cubic polynomial $ f(x) = ax^3 + bx^2 + cx + d $. Given the values $ f(1) = 15 $, $ f(2) = 30 $, $ f(3) = 45 $, and $ f(4) = 60 $, we aim to determine $ f(5) $—a key insight for future resource planning.", "---", "### Step 1: Observing the Pattern\nThe given data suggests a linear trend:\n- $ f(1) = 15 $\n- $ f(2) = 30 $ (−15 increase)\n- $ f(3) = 45 $ (−15 increase)\n- $ f(4) = 60 $ (−15 increase)", "Each additional year adds 15 visits, indicating a constant rate of change. However, since the model is assumed to be cubic, we look for a cubic function passing through these four points.", "---", "### Step 2: Setting Up the System of Equations\nWe substitute $ x = 1, 2, 3, 4 $ into $ f(x) = ax^3 + bx^2 + cx + d $ to form a system:", "$$\n\begin{aligned}\nf(1) &= a(1)^3 + b(1)^2 + c(1) + d = a + b + c + d = 15 \quad \ ext{(1)} \\nf(2) &= a(8) + b(4) + c(2) + d = 8a + 4b + 2c + d = 30 \quad \ ext{(2)} \\nf(3) &= a(27) + b(9) + c(3) + d = 27a + 9b + 3c + d = 45 \quad \ ext{(3)} \\nf(4) &= a(64) + b(16) + c(4) + d = 64a + 16b + 4c + d = 60 \quad \ ext{(4)}\n\end{aligned}\n$$", "---", "### Step 3: Solving the System\nSubtract (1) from (2):\n$$\n(8a + 4b + 2c + d) - (a + b + c + d) = 30 - 15 \Rightarrow 7a + 3b + c = 15 \quad \ ext{(5)}\n$$", "Subtract (2) from (3):\n$$\n(27a + 9b + 3c + d) - (8a + 4b + 2c + d) = 45 - 30 \Rightarrow 19a + 5b + c = 15 \quad \ ext{(6)}\n$$", "Subtract (3) from (4):\n$$\n(64a + 16b + 4c + d) - (27a + 9b + 3c + d) = 60 - 45 \Rightarrow 37a + 7b + c = 15 \quad \ ext{(7)}\n$$", "Subtract (5) from (6):\n$$\n(19a + 5b + c) - (7a + 3b + c) = 15 - 15 \Rightarrow 12a + 2b = 0 \Rightarrow 6a + b = 0 \quad \ ext{(8)}\n$$", "Subtract (6) from (7):\n$$\n(37a + 7b + c) - (19a + 5b + c) = 15 - 15 \Rightarrow 18a + 2b = 0 \Rightarrow 9a + b = 0 \quad \ ext{(9)}\n$$", "Subtract (8) from (9):\n$$\n(9a + b) - (6a + b) = 0 - 0 \Rightarrow 3a = 0 \Rightarrow a = 0\n$$", "Then from (8): $ 6(0) + b = 0 \Rightarrow b = 0 $\nFrom (5): $ 7(0) + 3(0) + c = 15 \Rightarrow c = 15 $\nFrom (1): $ 0 + 0 + 15 + d = 15 \Rightarrow d = 0 $", "---", "### Step 4: Final Function\nThus, the cubic function simplifies to linear:\n$$\nf(x) = 15x\n$$", "This confirms the earlier linear trend:\n- $ f(1) = 15 $\n- $ f(2) = 30 $\n- $ f(3) = 45 $\n- $ f(4) = 60 $\n- $ f(5) = 15 \ imes 5 = 75 $", "---", "### Step 5: Public Health Implication\nModeling prenatal visits as $ f(x) = 15x $ allows clinics to forecast patient load accurately. This linear pattern, though simple, supports proactive staffing and resource allocation. While real-world maternal health data may include nonlinear trends, cubic models like this help identify deviations from expected growth and inform targeted interventions.", "---", "Conclusion\nUsing a cubic function to analyze maternal health data, we found $ f(5) = 75 $, reinforcing the importance of predictive modeling in public health. As seen in Vancouver clinics, consistent tracking and modeling enable better planning—ultimately improving maternal and child health outcomes.", "Keywords:\nmaternal health, prenatal visits, cubic model, public health data analysis, Vancouver clinics, predictive modeling, public health communication, f(x) = 15x, healthcare analytics", "---", "Call to Action:\nPublic health communicators should leverage data modeling tools to uncover trends like this—empowering communities with evidence-based insights for healthier futures."]