Question: An underwater robot’s depth $ d(t) $ (in meters) satisfies $ d(t) = pt^3 + qt^2 + rt + s $. Given $ d(1) = 10 $, $ d'(1) = 12 $, $ d(2) = 28 $, and $ d'(2) = 30 $, find $ d(0) $.

["Title: Solving for Initial Depth: An Underwater Robot’s Depth Function Explained", "Meta Description:\nExplore how to determine the initial depth of an underwater robot using its depth function $ d(t) = pt^3 + qt^2 + rt + s $, with given conditions on depth and velocity at $ t = 1 $ and $ t = 2 $. Find $ d(0) $ step-by-step.", "---", "### Introduction\nUnderstanding the precise movement and position of underwater robots is crucial for mapping, exploration, and scientific missions. When the depth $ d(t) $ of such a robot is modeled by a cubic polynomial—$ d(t) = pt^3 + qt^2 + rt + s $—mathematicians and engineers use initial conditions to fully determine the function and predict behavior. In this article, we solve a practical problem: given four key conditions on depth and the derivative at two time points, we find the initial depth $ d(0) $.", "---", "### The Problem\nWe are given:\n- $ d(t) = pt^3 + qt^2 + rt + s $\n- $ d(1) = 10 $\n- $ d'(1) = 12 $\n- $ d(2) = 28 $\n- $ d'(2) = 30 $", "We are to compute $ d(0) $.", "---", "### Step 1: Express $ d(t) $ and its Derivative", "The depth function is:\n$$\nd(t) = pt^3 + qt^2 + rt + s\n$$", "Its derivative, representing the robot’s vertical velocity, is:\n$$\nd'(t) = 3pt^2 + 2qt + r\n$$", "---", "### Step 2: Apply Given Conditions", "#### Use $ d(1) = 10 $:\n$$\nd(1) = p(1)^3 + q(1)^2 + r(1) + s = p + q + r + s = 10 \quad \ ext{(Equation 1)}\n$$", "#### Use $ d'(1) = 12 $:\n$$\nd'(1) = 3p(1)^2 + 2q(1) + r = 3p + 2q + r = 12 \quad \ ext{(Equation 2)}\n$$", "#### Use $ d(2) = 28 $:\n$$\nd(2) = p(8) + q(4) + r(2) + s = 8p + 4q + 2r + s = 28 \quad \ ext{(Equation 3)}\n$$", "#### Use $ d'(2) = 30 $:\n$$\nd'(2) = 3p(4) + 2q(2) + r = 12p + 4q + r = 30 \quad \ ext{(Equation 4)}\n$$", "---", "### Step 3: Solve the System of Equations", "We now solve Equations 1–4 simultaneously.", "Equations:\n1. $ p + q + r + s = 10 $\n2. $ 3p + 2q + r = 12 $\n3. $ 8p + 4q + 2r + s = 28 $\n4. $ 12p + 4q + r = 30 $", "#### Step 3a: Eliminate $ r $ using Equations 2 and 4", "From Equation 2:\n$$\nr = 12 - 3p - 2q \quad \ ext{(Eq A)}\n$$", "Substitute into Equation 4:\n$$\n12p + 4q + (12 - 3p - 2q) = 30\n\Rightarrow 9p + 2q + 12 = 30\n\Rightarrow 9p + 2q = 18 \quad \ ext{(Eq B)}\n$$", "#### Step 3b: Eliminate $ r $ using Equations 1 and unknown $ s $", "We’ll eliminate $ s $ using Equations 1 and 3.\nFrom Equation 1:\n$$\ns = 10 - p - q - r\n$$", "Substitute $ r $ from Eq A:\n$$\ns = 10 - p - q - (12 - 3p - 2q) = 10 - p - q -12 + 3p + 2q = (2p + q - 2)\n\quad \ ext{(Eq C)}\n$$", "Now plug Eq C into Equation 3:\n$$\n8p + 4q + 2r + s = 28\n\Rightarrow 8p + 4q + 2(12 - 3p - 2q) + (2p + q - 2) = 28\n$$", "Expand:\n$$\n8p + 4q + 24 - 6p - 4q + 2p + q - 2 = 28\n\Rightarrow (8p - 6p + 2p) + (4q - 4q + q) + (24 - 2) = 28\n\Rightarrow 4p + q + 22 = 28\n\Rightarrow 4p + q = 6 \quad \ ext{(Eq D)}\n$$", "---", "### Step 4: Solve for $ p $ and $ q $", "We now solve:\n- $ 9p + 2q = 18 $ (Eq B)\n- $ 4p + q = 6 $  (Eq D)", "From Eq D:\n$$\nq = 6 - 4p\n$$", "Substitute into Eq B:\n$$\n9p + 2(6 - 4p) = 18\n\Rightarrow 9p + 12 - 8p = 18\n\Rightarrow p + 12 = 18\n\Rightarrow p = 6\n$$", "Now find $ q $:\n$$\nq = 6 - 4(6) = 6 - 24 = -18\n$$", "Find $ r $ from Eq A:\n$$\nr = 12 - 3(6) - 2(-18) = 12 - 18 + 36 = 30\n$$", "Find $ s $ from Eq C:\n$$\ns = 2(6) + (-18) - 2 = 12 - 18 - 2 = -8\n$$", "---", "### Step 5: Compute $ d(0) $", "By definition,\n$$\nd(0) = p(0)^3 + q(0)^2 + r(0) + s = s = -8\n$$", "---", "### Conclusion", "The underwater robot’s initial depth at $ t = 0 $ is $ d(0) = -8 $ meters. While depth below sea level is typically non-negative, the mathematical model correctly reflects its starting position relative to a chosen reference. Given the cubic model and the data, $ d(0) = -8 $ is the precise result.", "---", "Keywords: underwater robot depth function, $ d(t) = pt^3 + qt^2 + rt + s $, solve cubic polynomial conditions, underwater robotics math, $ d(0) $, mathematical modeling, depth derivative analysis", "References:\n- Algebraic systems involving polynomial interpolation\n- Ordinary differential equations in motion modeling\n- Application of initial conditions in engineering design", "---", "Ready to interpret underwater robot trajectories? Use this method to validate models and predict starting depths from real-world measurements."]