Math 150A Writing Project ( calculus 1 )

Anonymous
timer Asked: Jun 14th, 2018
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Question description

You must complete one of the following problems and produce a two-page write-up of the problem (in your own words), the full solution (withjustification of each step of the process and any conclusions drawn), and interpretation of the results. Any variables must clearly be defined. Give answers in fractional form (not decimal), if necessary.

  • At least 2 FULL pages, typed, double-spaced, 12 point font, 1 inch margins.
  • No grammatical, punctuation, or spelling errors.
  • Your name and title should be included in the header of the document and not the body.
  • Use the sample template.
  • Figures may be included on a separate page at the end, which will not be countedtowards the 2 full pages.
  • DO THE SECOND PROBLEM.

Math 150A Ladder Around a Corner Laura Smith, Ph.D. We are to carry a stepladder down a hallway that is 9 feet wide. Unfortunately, at the end of the hallway there is a corner, where we must make a right-angled turn into a hallway that is 6 feet wide. Our goal is to determine the maximum length of a ladder that could be carried horizontally around this corner. In order to solve this problem, we must consider a ladder that touches the exterior wall in our hallway, the exterior wall in the other hallway, and the interior corner. Any ladder that cannot touch all three will be short enough to pass. Thus, the problem has turned into finding the shortest ladder that will touch all three. Let L be the length of a ladder that touches the exterior wall in the 9-foot hallway at point A, the exterior wall in the 6-foot hallway at point B, and the interior corner at point C (see Figure 1). Extending the walls of the hallways on a diagram, we let D be the point where the 6-foot hallway extension intersects the wall, and let E be the point where the 9foot hallway extension intersects the other wall (see Figure 1). Let x be the length of line segment AC, and let y be the length of line segment CB. We notice that the angle  between line segments DC and AD is the same as the angle between line segments EB and BC. Using the definitions of trigonometric functions, we can write the length of the ladder L as a function of this angle , 𝐿(πœƒ) = π‘₯ + 𝑦 = 9 βˆ™ csc(πœƒ) + 6 βˆ™ sec(πœƒ). πœ‹ The domain of this function is 0 < πœƒ < 2 . To find the minimal L, we find our absolute minimum using the first derivative test for absolute extreme values. Thus, we need to find the critical numbers by taking the derivative. We have 𝐿′ (πœƒ) = βˆ’9 βˆ™ csc(πœƒ) βˆ™ cot(πœƒ) + 6 βˆ™ sec(πœƒ) βˆ™ tan(πœƒ). To find the critical numbers, we set our derivative equal to zero. This gives us 0= βˆ’9cos(πœƒ) sin2 (πœƒ) + 6sin(πœƒ) cos2 (πœƒ) = βˆ’9 cos3(πœƒ)+6sin3 (πœƒ) sin2 (πœƒ)cos2 (πœƒ) , which is zero when the numerator is equal to zero. Hence, βˆ’9 cos 3 (πœƒ) + 6sin3 (πœƒ) = 0, or 3 equivalently, 2 = π‘‘π‘Žπ‘›3 (πœƒ). Thus, one critical number is 3 1/3 πœƒ = π‘‘π‘Žπ‘›βˆ’1 ((2) ). We then search for when the derivative is undefined, which occurs when either sine or cosine is equal to zero on the domain. However, there are no such values on our domain. We now examine where L’() is positive and negative on our domain. We have that 3 1/3 our derivative is negative for 0<<π‘‘π‘Žπ‘›βˆ’1 ((2) 3 1/3 ) and positive for π‘‘π‘Žπ‘›βˆ’1 ((2) πœ‹ ) < πœƒ < 2. According to the first derivative test for absolute extreme values, we must have an absolute 3 1/3 minimum at πœƒ = π‘‘π‘Žπ‘›βˆ’1 ((2) ). Thus, the minimum length of a ladder that touches the points A, B, and C has length 1 1 3 3 3 3 𝐿(πœƒ) = 9 βˆ™ csc (π‘‘π‘Žπ‘›βˆ’1 (( ) )) + 6 βˆ™ sec (π‘‘π‘Žπ‘›βˆ’1 (( ) )) feet β‰ˆ 21.1 feet . 2 2 By the equivalence of the problems, we must have that the longest ladder that could be 1 carried horizontally around this corner is 9 csc (π‘‘π‘Žπ‘› βˆ’1 3 3 ((2) )) 1 + 6 sec (π‘‘π‘Žπ‘› βˆ’1 3 3 ((2) )) feet, or approximately 21.1 feet. In examining a few stores that sell ladders, the longest stepladders sold are typically 20 feet. Therefore, we would be able to successfully carry our stepladder around the corner. However, extension ladders are typically longer and might not make the corner. 9-Foot Hallway A x D ΞΈ C 9 y 6 E ΞΈ 6-Foot Hallway B Figure 1. The black lines indicate the walls of the hallways. The points A and B are where the ladder touches the exterior walls. The point C is where the ladder touches the interior corner. Points D and E are where the wall extensions would intersect the exterior walls.
Math 150A Writing Project In order to satisfy the GE university writing requirements for the course, you are to complete writing project. You must complete one of the following problems and produce a two-page write-up of the problem (in your own words), the full solution (with justification of each step of the process and any conclusions drawn), and interpretation of the results. Any variables must clearly be defined. Give answers in fractional form (not decimal), if necessary. Problem 1 – Filling Swimming Pools: Spring in Southern California has finally arrived. You are looking forward to filling your swimming pool, which has a length of 10 meters and ends with trapezoidal shape. The trapezoids on the ends have a bottom edge with length 6 meters, a top edge with length 12 meters, and a height of 2 meters. Your hose can produce 3 cubic meters of water per hour. You begin to fill your pool. When the height of the water reaches 0.1 m, you realize this will take a long time. Calculate the rate the height of the water is increasing when it is 0.1 m high. Use this value to estimate how long it will take to fill the pool. Is this an overestimate or underestimate? Explain your reasoning. Problem 2 – Party Games: Setting up for a birthday party, you decide to play a game with party hats. The hats are in the shape of a cone with height 10 inches and radius 6 inches. The hats are placed on a table with the point towards the ceiling. The goal of the game is guess the correct hat with a prize hidden underneath it. The prize is contained in a cylindrical cup. You need to determine the dimensions of the cylindrical cup (radius and height) that can hold the most and still be hidden under a party hat. What is the most that this optimal cup can hold? Given this information, what would you include as the prize? Requirements: β€’ β€’ β€’ β€’ β€’ At least 2 FULL pages, typed, double-spaced, 12 point font, 1 inch margins. No grammatical, punctuation, or spelling errors. Your name and title should be included in the header of the document and not the body. Use the sample template. Figures may be included on a separate page at the end, which will not be counted towards the 2 full pages. Academic Dishonesty I will pursue these cases. Do not plagiarize. If you work with another person, be sure to produce the write-up independently. Due Thursday, June 28, 2018 at 8:00 A.M.

Tutor Answer

Borys S
School: University of Virginia

Well, I wrote even 2+ pages, please review the result. I believe you'll have some remarks and/or wishes. We still have enough time to implement them.

Math 150A

Party Games

NAME

Consider our hat that has a form of a cone, and a prize inside it that has a form of
a cylinder. Or to be stricter, they are right circular cone and right circular cylinder.
Suppose that the cylinder stays on its base inside the cylinder. Later we'll consider
the opposite option when it stays on its side surface. Then make a section of them both
with a vertical plane that goes through the vertex of the hat and the vertical axis of the
cylinder. Of course, such a plane goes through the axis of the cone, too.
Suppose also that the cylinder touches the side surface of the cone by the cylinder's
top circumference, s...

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Review

Anonymous
Wow this is really good.... didn't expect it. Sweet!!!!

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