Week 8 Assignment: Signature Assignment (1) A Skier Is Tryin ✓ Solved

Week 8 Assignment: Signature Assignment (1) A skier is tryin

Week 8 Assignment: Signature Assignment (1) A skier is trying to decide whether or not to buy a season ski pass. A daily pass costs $67. A season ski pass costs $350. The skier would have to rent skis with either pass for $25 per day. Using linear equations, explain the process to find how many ski days it would take the skier to make the season pass less expensive than daily passes.

(2) Ace Manufacturing has determined that the cost of labor for producing transmissions is dollars, while the cost of materials is dollars. a) Write a polynomial that represents the total cost of materials and labor for producing transmissions. b) Using the website, graph the polynomial, using the settings 0 to 1000 (step 100) for the x-axis and 0 to 1,000,000 (step 100,000) for the y-axis. Do not use commas in any of the numbers. Share the graph by exporting and downloading the image. Include the graph as part of your solution. c) Using the function and the graph, explain the process to find the total cost for producing 500 transmissions. d) Verify the answer from the previous problem by finding the cost of labor and cost of materials for producing 500 transmissions.

(3) The area of a rectangle is modeled by the polynomial square units. The width of the rectangle is modeled by the polynomial units. Showing all work, write a polynomial that represents the length of the rectangle.

(4) In 2012, Hurricane Sandy struck the East Coast of the United States, killing 147 people and causing an estimated $75 billion dollars in damage. With a gale diameter of about 1,000 miles, it was the largest ever to form over the Atlantic Basin. The accompanying data represent the number of major hurricane strikes in the Atlantic Basin (category 3, 4, or 5) each decade from 1921 to 2010. Decade, Major Atlantic Basin Hurricanes 1921–1930, –1940, –1950, –1960, –1970, –1980, –1990, –2000, –2010, The polynomial function models the number of major Atlantic Basin hurricanes from the time period given in the table above. a) Using the website, graph the polynomial using the settings 0 to 10 (step 1) for the x-axis and 0 to 40 (step 5) for the y-axis. Share the graph by exporting and downloading the image. Include the graph as part of your solution. b) Using the model and showing all work, estimate the number of major Atlantic Basin hurricanes between 1961 and 1970. Does this overestimate or underestimate the actual data shown in the table? By how much? c) Using the model and showing all work, predict the number of major Atlantic Basin hurricanes between 2011 and 2020. Is your result reasonable? d) The graph shows periods of increasing and periods of decreasing. Based on the graph alone, around what year did the number of major Atlantic Basin hurricanes start to decrease? Explain. Around what year did the number of major Atlantic Basin hurricanes begin to increase again? Explain.

(5) A balloonist throws a sandbag downward at 24 feet per second from an altitude of 720 feet. Its height (in feet) above the ground after seconds is given by . a) What is the height of the sandbag 2 seconds after it is thrown? Explain. b) How long does it take for the sandbag to reach the ground? Explain.

(6) A drug injected into a patient and the concentration of the drug in the bloodstream is monitored. The drug’s concentration, , in milligrams per liter, after hours is modeled by . a) Use the site to draw the graph of this function. Once you type the function in the cell (upper left), click the wrench symbol in the upper right (graph settings) and make the -axis from 0 to 10 (step 1) and the -axis from 0 to 3 (step 1). Share the graph by exporting and downloading the image. Include the graph as part of your discussion. b) Using the graph, explain what happens to the drug concentration in the bloodstream over time. c) Explaining the process, find the drug’s concentration after 3 hours. d) What is the highest concentration of the drug in the bloodstream and when does it take place? Explain.

(7) The sonar of a Navy cruiser detects a submarine that is 2500 feet away. The point on the water directly above the submarine is 1500 feet away from the cruiser. What is the depth of the submarine? Explain.

(8) The polynomial function models a man’s normal systolic blood pressure, , at age . Showing all work, find the age, to the nearest year, of a man whose normal blood pressure is 125 mm Hg. Explain why one of the solutions is not relevant.

Paper For Above Instructions

The following paper provides solutions to the Week 8 Assignment questions (1)–(8) using standard algebraic techniques, linear models, and polynomial reasoning. Throughout, a mix of symbolic expressions and numerical examples is used to illustrate procedure, with attention to units, interpretation, and verification. The content integrates general methods for breaking-even analysis, cost modeling, polynomial operations, and basic kinematic and biological modeling. See the references for detailed explanations of the mathematical principles and graphing practices used below (Khan Academy; Paul’s Online Math Notes; OpenStax; Investopedia; Britannica).

1) Ski pass break-even analysis. Let d be the number of skiing days. Daily pass cost: 67d. Ski rental per day: 25d. Season pass base cost: 350 plus same rental per day: 25d. The total cost with a daily-pass regime is 67d + 25d = 92d. The total cost with the season pass is 350 + 25d. Break-even occurs when 92d = 350 + 25d, so 67d = 350 and d = 350/67 ≈ 5.22 days. Therefore, for any d ≥ 6 days, the season pass becomes less expensive than paying daily. This is a classic break-even problem in cost accounting, and the approach—set the two cost expressions equal and solve for d—is standard in many introductory algebra and business math resources (Investopedia, 2023; Khan Academy, n.d.).

2) Transmission cost polynomial. Let L denote the cost of labor per transmission and M denote the cost of materials per transmission, with F representing any fixed cost. If n transmissions are produced, the total cost is C(n) = F + (L + M)n, a linear polynomial in n. If a fixed charge F is zero, C(n) reduces to C(n) = (L + M)n. The graph of this linear function can be constructed by plotting C(n) on the y-axis against n on the x-axis, using the specified domain 0 to 1000 and range 0 to 1,000,000. In the absence of explicit numeric values for L, M, and F, the expression above is the appropriate general form (OpenStax, Algebra & Trigonometry; Khan Academy; Paul’s Notes). To find the total cost for 500 transmissions, substitute n = 500: C(500) = F + (L + M)·500. Labor cost for 500 units equals L·500, and materials cost equals M·500. These components align with standard cost-accounting practice where total cost is the sum of fixed costs plus variable costs per unit (Investopedia, 2023).

3) Area/width/length polynomial. If A(x) denotes the area polynomial and W(x) denotes the width polynomial, then the length L(x) satisfies A(x) = W(x)·L(x). Solving for L(x) yields L(x) = A(x)/W(x). This quotient represents a polynomial divided by a polynomial; in general, the result is a rational function unless W(x) divides A(x) exactly. This aligns with polynomial algebra coursework on division of polynomials and the interpretation of area as a product of length and width (OpenStax; Khan Academy; Paul's Notes).

4) Hurricane Sandy model and hurricane counts. To address this item, one would collect decade-based counts of major Atlantic Basin hurricanes (category 3–5) from 1921 to 2010 and fit a polynomial model, typically of degree 2 or 3, using x-values that encode decade indices (e.g., 0 for 1921–1930, 1 for 1931–1940, etc.). Steps include graphing the polynomial with the specified domain and range, using least-squares regression or interpolation. Then, evaluate the model for 1961–1970 (which corresponds to a particular x-value), compare the estimate to the tabulated data to determine over- or underestimation, and quantify the difference. Next, predict for 2011–2020 by evaluating the model at the corresponding x-value and assess reasonableness by considering model assumptions and data limitations. Finally, inspect the graph to infer approximate years when the hurricane count began to decrease and when it began to rise again. This modeling approach relies on standard regression techniques and interpretation of polynomial fits in time-series data (Britannica; Khan Academy; OpenStax).

5) Sandbag height problem. With height h(t) given by a quadratic model h(t) = 720 − 24t − 16t^2 (assuming standard gravity g = 32 ft/s^2 so the second term is −24t and the acceleration term is −16t^2), the height two seconds after release is h(2) = 720 − 48 − 64 = 608 feet. To find when the sandbag hits the ground, solve h(t) = 0: 720 − 24t − 16t^2 = 0, or 16t^2 + 24t − 720 = 0. Dividing by 8 gives 2t^2 + 3t − 90 = 0, which yields t = [−3 + √(9 + 720)]/4 = [−3 + 27]/4 = 24/4 = 6 seconds (the negative root is non-physical). This illustrates using kinematic equations for vertical motion under constant acceleration (Khan Academy; OpenStax).

6) Drug concentration model. If the concentration C(t) follows a standard decay or production model such as C(t) = C0 e^{−kt} (or a more complex pharmacokinetic form provided by the problem), one would graph the function over t ∈ [0, 10] with appropriate axis settings as described, then interpret the behavior: typically C(t) decreases over time for elimination-dominated processes. The concentration after 3 hours is C(3) = C0 e^{−3k}. The maximum concentration occurs at t = 0 for a purely decaying model. If a more complex function is given (e.g., dosing regimens or absorption terms), the optimum time for peak concentration would be determined by derivative analysis dC/dt = 0, and the interpretation would depend on the actual function (Khan Academy; OpenStax; Investopedia).

7) Submarine depth from sonar geometry. The cruiser-submarine distance is 2500 ft, and the horizontal separation to the point above the submarine is 1500 ft. By the right-triangle relation, 2500^2 = 1500^2 + d^2, so d^2 = 2500^2 − 1500^2 = 6.25×10^6 − 2.25×10^6 = 4.0×10^6, giving d = 2000 ft. This is a straightforward application of the Pythagorean theorem in a right-triangle scenario (Britannica; Khan Academy).

8) Blood pressure model. With a polynomial P(age) modeling normal systolic blood pressure, solving P(age) = 125 yields potentially two mathematical solutions for age. Only ages within a plausible human lifespan (e.g., 0–120 years) are meaningful; one solution may be extraneous if the model uses even powers or squaring operations that introduce non-physical roots. The reasoning follows standard polynomial solving and model interpretation principles discussed in calculus and algebra texts (OpenStax; Khan Academy; Paul's Notes).

References

  • Khan Academy. (n.d.). Graphing polynomials. https://www.khanacademy.org/math/algebra/polynomials
  • Paul’s Online Math Notes. (n.d.). Polynomial Functions. http://tutorial.math.lamar.edu/Classes/Alg/PolynomialFunctions.aspx
  • OpenStax. (2017). Algebra and Trigonometry. OpenStax CNX. https://openstax.org/details/books/algebra-and-trigonometry
  • Investopedia. (2023). Cost function. https://www.investopedia.com/terms/c/cost-function.asp
  • Britannica. (n.d.). Polynomial. https://www.britannica.com/topic/polynomial
  • Wolfram Alpha. (n.d.). Graph a polynomial. https://www.wolframalpha.com/
  • Larson, R., & Edwards, B. (2013). Calculus: Early Transcendental Functions (6th ed.). Brooks/Cole.
  • Stewart, J. (2015). Calculus: Early Transcendentals (8th ed.). Cengage Learning.
  • Anton, H., Bivens, I., & Davis, S. (2012). Calculus: Early Transcendentals (9th ed.). Wiley.
  • OpenStax. (2020). Calculus. OpenStax. https://openstax.org/details/books/calculus

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