A mammalogist is studying the diving behavior of a species of dolphin. The depth \( D(t) \) in meters that a dolphin reaches after \( t \) seconds is modeled by \( D(t) = pt^3 + qt^2 + rt + s \). Given \( D(1) = 3 \), \( D(2) = 14 \), \( D(3) = 39 \), and \( D(4) = 84 \), find the coefficients \( p \), \( q \), \( r \), and \( s \).

["Unraveling Dolphin Diving Behavior: A Mammalogist’s Statistical Analysis", "Understanding how dolphins dive is crucial for marine biology and conservation. A recent study by a dedicated mammalogist uses a cubic model to describe the depth ( D(t) ) of a dolphin over time, revealing precise insights into its diving patterns. The depth is modeled as:", "[\nD(t) = pt^3 + qt^2 + rt + s\n]", "Using experimental data from four time points—( D(1) = 3 ), ( D(2) = 14 ), ( D(3) = 39 ), and ( D(4) = 84 )—the researcher determines the coefficients ( p ), ( q ), ( r ), and ( s ) to refine the model.", "### Setting Up the System of Equations", "Substitute each data point into the cubic equation:", "1. At ( t = 1 ):\n[\np(1)^3 + q(1)^2 + r(1) + s = 3 \Rightarrow p + q + r + s = 3\n]", "2. At ( t = 2 ):\n[\np(8) + q(4) + r(2) + s = 14 \Rightarrow 8p + 4q + 2r + s = 14\n]", "3. At ( t = 3 ):\n[\np(27) + q(9) + r(3) + s = 39 \Rightarrow 27p + 9q + 3r + s = 39\n]", "4. At ( t = 4 ):\n[\np(64) + q(16) + r(4) + s = 84 \Rightarrow 64p + 16q + 4r + s = 84\n]", "### Solving the System Step-by-Step", "We now solve the system:", "[\n\begin{cases}\n(1)\quad p + q + r + s = 3 \\n(2)\quad 8p + 4q + 2r + s = 14 \\n(3)\quad 27p + 9q + 3r + s = 39 \\n(4)\quad 64p + 16q + 4r + s = 84 \\n\end{cases}\n]", "Step 1: Eliminate ( s )", "Subtract equation (1) from (2):\n[\n(8p + 4q + 2r + s) - (p + q + r + s) = 14 - 3 \Rightarrow 7p + 3q + r = 11 \quad \ ext{(5)}\n]", "Subtract (2) from (3):\n[\n(27p + 9q + 3r + s) - (8p + 4q + 2r + s) = 39 - 14 \Rightarrow 19p + 5q + r = 25 \quad \ ext{(6)}\n]", "Subtract (3) from (4):\n[\n(64p + 16q + 4r + s) - (27p + 9q + 3r + s) = 84 - 39 \Rightarrow 37p + 7q + r = 45 \quad \ ext{(7)}\n]", "Step 2: Eliminate ( r )", "Subtract (5) from (6):\n[\n(19p + 5q + r) - (7p + 3q + r) = 25 - 11 \Rightarrow 12p + 2q = 14 \Rightarrow 6p + q = 7 \quad \ ext{(8)}\n]", "Subtract (6) from (7):\n[\n(37p + 7q + r) - (19p + 5q + r) = 45 - 25 \Rightarrow 18p + 2q = 20 \Rightarrow 9p + q = 10 \quad \ ext{(9)}\n]", "Step 3: Solve for ( p ) and ( q )", "Subtract (8) from (9):\n[\n(9p + q) - (6p + q) = 10 - 7 \Rightarrow 3p = 3 \Rightarrow p = 1\n]", "Substitute ( p = 1 ) into (8):\n[\n6(1) + q = 7 \Rightarrow q = 1\n]", "Step 4: Solve for ( r )", "Use equation (5) with ( p = 1 ), ( q = 1 ):\n[\n7(1) + 3(1) + r = 11 \Rightarrow 7 + 3 + r = 11 \Rightarrow r = 1\n]", "Step 5: Solve for ( s )", "Use equation (1):\n[\n1 + 1 + 1 + s = 3 \Rightarrow s = 0\n]", "### Final Coefficients", "The cubic model describing the dolphin’s diving depth is:", "[\nD(t) = t^3 + t^2 + t\n]", "Thus, the coefficients are:", "[\np = 1,\quad q = 1,\quad r = 1,\quad s = 0\n]", "### Conclusion", "By analyzing real-world data through mathematical modeling, the mammalogist has successfully derived a precise cubic function that captures the dolphin’s diving behavior. This analysis not only validates the model’s accuracy but also supports deeper ecological studies and informed conservation strategies. Understanding these patterns helps protect marine life and enhances our knowledge of cetacean behavior.", "---", "Keywords: mammalogist, dolphin diving, cubic model, ( D(t) = pt^3 + qt^2 + rt + s ), depth function, marine biology, data analysis, cubic regression, marine mammal research."]