An entomologist is studying insect pollination and models the number of flowers \( F \) pollinated by bees as a quadratic function of time \( t \) in hours: \( F(t) = pt^2 + qt + r \). If 15 flowers are pollinated after 1 hour, 40 after 2 hours, and 75 after 3 hours, determine the values of \( p \), \( q \), and \( r \).

["Title: Solving for the Pollination Model: How an Entomologist Unraveled Bee Flower Pollination Rates", "Meta Description:\nAn entomologist models bee pollination using a quadratic function ( F(t) = pt^2 + qt + r ). Using data from 1, 2, and 3 hours, we determine the coefficients ( p ), ( q ), and ( r ) to accurately predict flower pollination.", "---", "### Introduction to the Study", "In the field of entomology, understanding pollination dynamics is essential for ecosystem health and agriculture. An expert entomologist recently set out to model how quickly bees pollinate flowers over time. Using precise data collected from hourly observations, they formulated a quadratic equation to describe the cumulative number of flowers pollinated:", "[\nF(t) = pt^2 + qt + r\n]", "Here, ( F(t) ) represents the total number of flowers pollinated by bees after ( t ) hours. To build an accurate model, the entomologist used three key data points:\n- ( F(1) = 15 )\n- ( F(2) = 40 )\n- ( F(3) = 75 )", "These values allowed solving for the unknown coefficients ( p ), ( q ), and ( r )—critical parameters that reveal the pattern of pollination growth over time.", "---", "### Setting Up the System of Equations", "From the model ( F(t) = pt^2 + qt + r ), we substitute the known data points to form a system of equations.", "1. At ( t = 1 ):\n[\np(1)^2 + q(1) + r = 15 \quad \Rightarrow \quad p + q + r = 15 \ ag{1}\n]", "2. At ( t = 2 ):\n[\np(2)^2 + q(2) + r = 40 \quad \Rightarrow \quad 4p + 2q + r = 40 \ ag{2}\n]", "3. At ( t = 3 ):\n[\np(3)^2 + q(3) + r = 75 \quad \Rightarrow \quad 9p + 3q + r = 75 \ ag{3}\n]", "Now, solve this system of three equations simultaneously.", "---", "### Solving the System Step-by-Step", "Step 1: Subtract Equation (1) from Equation (2)\n[\n(4p + 2q + r) - (p + q + r) = 40 - 15 \Rightarrow 3p + q = 25 \ ag{4}\n]", "Step 2: Subtract Equation (2) from Equation (3)\n[\n(9p + 3q + r) - (4p + 2q + r) = 75 - 40 \Rightarrow 5p + q = 35 \ ag{5}\n]", "Step 3: Subtract Equation (4) from Equation (5)\n[\n(5p + q) - (3p + q) = 35 - 25 \Rightarrow 2p = 10 \Rightarrow p = 5\n]", "Step 4: Substitute ( p = 5 ) into Equation (4)\n[\n3(5) + q = 25 \Rightarrow 15 + q = 25 \Rightarrow q = 10\n]", "Step 5: Substitute ( p = 5 ), ( q = 10 ) into Equation (1)\n[\n5 + 10 + r = 15 \Rightarrow r = 0\n]", "---", "### Final Results", "The entomologist’s quadratic model for flower pollination is:\n[\nF(t) = 5t^2 + 10t + 0 = 5t^2 + 10t\n]", "Thus, the coefficients are:\n- ( p = 5 )\n- ( q = 10 )\n- ( r = 0 )", "This means the number of flowers pollinated grows quadratically, accelerating over time—consistent with bees increasing their foraging activity.", "---", "### Conclusion", "By combining precise field data with algebraic modeling, the entomologist established a reliable equation that quantifies bee pollination rates. Understanding such patterns helps conservation efforts, improves crop pollination strategies, and deepens our knowledge of pollinator behavior.", "For researchers and scientists, this example highlights how small, measurable data points can unlock powerful insights into natural processes—powered by models rooted in real-world observation.", "Keywords: entomologist, insect pollination, quadratic model, F(t) = pt² + qt + r, bee pollination, pollination data, agricultural science, plant-insect interaction."]