Directions: Please help with the following questions in the image!

Please provide WORK for my understanding for all the questions! ALL REFERENCES AND ETC IS ALREADY ATTACHED PLEASE!

Note: THIS ISNT FOR AN ASSIGNMENT BUT MERELY FOR MY OWN UNDERSTANDING AND COMPREHENSION! PLEASE BE ORGINAL! THIS IS FOR PRACTICE.

An ice cube is freezing in such a way that the side length s, in inches, is s() = -t+6, where t is in hours. The surface area N of the ice cube is the function A(s) = 6s2. Part A: Write an equation that gives the volume at t hours after freezing begins. Part B: Find the surface area as a function of time, using composition, and determine its range. Part C: After how many hours will the surface area equal 384 square inches? Show all necessary calculations. .Many doctors rely on the use of intravenous medication administration in order to achieve an immediate response of a particular drug's effects. The concentration, C, in mg/L, of a particular medication after being injected into a patient can be given by the function of - -181 +541 where the time, t is hours after injection. 12 + 71 + 10 Part A: What is the domain of the function C(t) based on the context of the problem? Show all necessary calculations. Part B: Graph the function to determine the greatest concentration of the medication that a patient will have in their body.A railroad's track can be determined using the following graph. 413-944-6444; Several different roadways are in the same region as the railroad. Part A: A highway's path can be found using the equation 2):; + By = 21. Use the graphs of the functions to determine the number of intersections there will be between the railroad and the highway, and explain completely. Part E: A turnpike's route is determined by the equation 1; = if. Prove algebraically how many intersections there will be between the railroad and the turnpike, showing all necessaryr work. A polynomial f (x) has the given zeros of 6, -1, and -3. Part A: Using the Factor Theorem, determine the polynomial f (x) in expanded form. Show all necessary calculations. Part B: Divide the polynomial f (x) by (x- -x - 2) to create a rational function g(x) in simplest factored form. Determine g(x) and find its slant asymptote. Part C: List all locations and types of discontinuities of the function g(x).A right triangle is shown in the graph. y (x, y) (a, b) b Part A: Use the Pythagorean Theorem to derive the standard equation of the circle, with center at (a, b) and a point on the circle at (x, y). Show all necessary math work. Part B: If (a, b) = (5, -2) and c = 10, determine the domain and range of the circle. Part C: Is the point (10, 2) inside the border of the circle if (a, b) = (5, -2) and c = 10? Explain using mathematical evidence.A jumping spider's movement is modeled by a parabola. The spider makes a single jump from the origin and reaches a maximum height of 16 mm halfway across a horizontal distance of 80 mm. Part A: Write the equation of the parabola in standard form that models the spider's jump. Show your work. Part B: Identify the focus, directrix, and axis of symmetry of the parabola