Polynomial Technology Worksheets Grade 12

1. Sophia is an environmental engineer modeling the water flow rate in a river after a storm. The flow rate F(t) in thousands of cubic meters per hour is given by the polynomial function F(t) = -0.02t³ + 0.48t² - 2.88t + 6.4, where t is the time in hours after the storm begins (t ≥ 0). Environmental regulations require that for safety, the flow rate must be decreasing and must remain below 3.2 thousand cubic meters per hour. Using a graphing calculator or Desmos, determine the time interval(s) during which the river flow rate meets both safety conditions.
2. Kaia uses a graphing calculator to analyze the polynomial f(x) = -3x^5 + 15x^3 - 12x. She observes the graph and notes the x-intercepts and the behavior around them. Using the calculator, determine all real zeros of the polynomial and classify each zero as having odd or even multiplicity based on whether the graph crosses or touches the x-axis at that point. Then, write the polynomial in completely factored form over the real numbers.
3. A biomedical engineer is modeling the concentration of a new medication in a patient's bloodstream using the polynomial function C(t) = -0.02t³ + 0.36t² - 1.8t + 2.4, where C(t) represents the concentration in milligrams per liter and t is the time in hours after administration. The medication becomes effective when the concentration reaches its maximum value. At what time does the medication become effective, and what is the maximum concentration?

…and 4 more problems

1. Liam is designing a roller coaster track that follows the path of the polynomial function f(x) = 2x³ - 9x² - 24x + 5. He needs to determine the exact coordinates of all local extrema points to ensure proper safety engineering. Find all critical points where the roller coaster reaches its highest and lowest points along this path.
2. A biomedical engineer is modeling the concentration of a new drug in a patient's bloodstream using the polynomial function C(t) = -0.02t³ + 0.3t² + 0.5t, where C is the concentration in milligrams per liter and t is time in hours after administration. The drug is considered effective when the concentration is increasing and above 1 mg/L. Determine the time interval during which the drug remains effective.
3. An environmental engineer is modeling the population growth of an endangered species in a wildlife reserve using the polynomial function P(t) = -0.01t³ + 0.15t² + 0.6t + 20, where P(t) represents the population in hundreds and t is the time in years since conservation efforts began. The species is considered to be recovering when the population growth rate is positive and the population exceeds 2,500 individuals. During what time interval is the species considered to be recovering?

…and 3 more problems

1. A polynomial function f(x) is graphed on a coordinate plane. The graph shows the function crossing the x-axis at (-4, 0) and (2, 0), and touching but not crossing the x-axis at (-1, 0). The graph passes through the point (1, -12). The end behavior shows that as x → -∞, f(x) → -∞ and as x → ∞, f(x) → ∞. Determine the equation of this polynomial in standard form.
2. Use Desmos to graph f(x)=x⁴-8x³+22x²-24x+8 and find the sum of all x-coordinates where the graph has local minima.
3. An environmental engineer is modeling the concentration of a pollutant in a river system using the polynomial function P(t) = -0.01t⁴ + 0.12t³ + 0.2t² - 1.2t + 2, where P(t) represents the pollutant concentration in parts per million and t is the time in days after a chemical spill. Environmental regulations require the pollutant concentration to remain below 1.5 ppm. Determine the time intervals during which the pollutant concentration exceeds the regulatory limit.

…and 4 more problems