Written Assignment Unit 2 PDF

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Week two assignment from written assignment unit 2 from academic year 2021-2022...

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Written Assignment Unit 2 Questions For this written assignment, answer the following questions showing all of your work. 1. Determine whether the lines given by the equations below are parallel, perpendicular, or neither. Also, find a rigorous algebraic solution for each problem. a.

b.

c.

[Suggestion: go to www.desmos.com/calculator, write the two equations and try to conclude the answer.] Answers a) 3y + 4x = 12, hence, slope = b / -2a => slope = -4/3 -6y = 8x + 1, hence, slope = -8/6 => slope = -4/3. The slope are the same, hence it is parallel. Graph is as follows:

b) 3y + x = 12, has slope = -1/3, y = 8x + 1 has slope 8. Hence, the slope are different, neither parallel nor perpendicular. There will intersects at a point. Graph is as follows:

c) 4x – 7y = 10, has slope = 4/7, 7x + 4y = 1 has slope = -7/4. The slope are negative reciprocals, so the line are perpendicular.

Graph is as follows:

Questions 2. A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by . What is the height of the building? What is the maximum height reached by the ball? How long does it take to reach maximum height? Also, find a rigorous algebraic solution for the problem. [Suggestion: go to www.desmos.com/calculator and write

and observe the answers.] Answers Since h(t) = -4.9t2 + 24t + 8, at the top before throwing, the t = 0 the height is 8 meters. For maximum height, we need to differentiate the given function with regard to the time and put it to 0. Hence, the differentiation are: 2 d (h ( t ) ) d (−4.9 t + 24 t+8) = dt dt 2

=

d (−4.9 t ) d ( 24 t ) d ( 8) + + dt dt dt

= - (2 * 4.9) t + 24 + 0 = -9.8t + 24 Since

d (h ( t ) ) =0 , hence, - 9.8t + 24 = 0, => t = 2.449, dt

The maximum time to reach the maximum height is at t = 2.449 sec. Put t into the function h (2.449) = -4.9 * 2.4492 + 24 * 2.449 + 8 = 37.388 Hence, the maximum height is 37.388 meters. The graphs is showing below:

Questions 3. A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest? Also, find a rigorous algebraic solution for the problem. [Suggestion: finding a function B(n), that is, the number of bushels corresponding to n trees per acre, go to the www.desmos.com/calculator and try to use a graph to visualize the situation and test your formula.] Answers Total brushes for 75 per acre = 75 * 20 = 1500 Since each tree decreases by 3 bushels, hence, y = (75 +x) * 17 This is bigger than 1500. Hence, 1500 < 1275 + 17x, 17x > 225, hence, x >= 13.235 since x can be integer only, hence, at least 14 trees must be added to each acre in addition to make it better deal for fruits. The graph is as follows:

References Abramson, J. (2017). Algebra and trigonometry. OpenStax, TX: Rice University. Retrieved from https://openstax.org/details/books/algebra-and-trigonometry Desmos. (2017, May 09). Desmos graphics calculator. https://www.desmos.com/calculator? lang=en...