Скачать или смотреть Derivation, Integration, and Series

Cost-optimized mailing tube, Wind load on a billboard, Iteration efficiency & series

F. Task
Task 1 — Cost-optimized mailing tube packaging A campus print shop needs a closed cylindrical mailing tube for A1 posters. The internal volume
is fixed at 𝑉 = 0.010m3
10 000 cm3. Since you have mastered the concept of Engineering
Mathematics I, you took the imitative to design this cylindrical mailing tube.
Material costs differ: curved sidewall costs 𝑐𝑠 = 𝑅𝑀 18/m2 ; end-caps cost 𝑐𝑒 = 𝑅𝑀 28/m2. You are required to write the cost of the cylindrical mailing tube as 𝐶𝑟, r is the radius of the cylindrical
mailing tube. Using the concept of differentiation such as 𝐶'𝑟 𝑎𝑛𝑑 𝐶′′𝑟 to find the stationary
point, r*. Show that this is the minimum. Then, you need to compute the optimal dimensions for the radius and height and the calculate the minimum cost 𝐶min for this cylindrical mailing tube.
Analyze when the end-caps cost rise to 𝑐𝑒 = 𝑅𝑀 35/m2
sidewall unchanged, explain how r*
shifts increase/decrease.

Task 2 - Wind load on a billboard
Behind the cylindrical mailing tube in Task 1, you have set up a rectangular billboard is mounted
vertically. Width 𝑏 = 5.0 m, height 𝐻 = 4.0 m. Idealize wind pressure as height-dependent is
given as:
𝑝𝑦 = 𝑝0 + 𝑘 𝑦, 0 ≤ 𝑦 ≤ 𝐻. H is the height measured from the bottom edge, and 𝑝0 = 40N/m2, 𝑘 = 12 N/m3
Evaluate the total force in N using:
𝐹 = ∫ 𝑝𝑦 𝑏𝐻0𝑑𝑦
Then, find the height of the resultant center of pressure, m, from the bottom edge using:
𝑦𝑐𝑝 =∫ 𝑦 𝑝𝑦 𝑏𝐻0𝑑𝑦/
∫ 𝑝𝑦 𝑏𝐻0𝑑𝑦
Explain whether 𝑦𝑐𝑝 is higher or lower, when 𝑝𝑦 = 𝑝0? Give a 2–3 sentence explanation.
Task 3 - Iteration efficiency & series
When optimizing the design of the cylindrical mailing tube in Task 1, you are trying to minimize
the errors for each optimization cycle. After each cycle of optimization, the design error is 60% of
the previous error. For instance, initial relative error 𝐸0 = 5.0%. After 𝑛 cycles: 𝐸𝑛 = 𝐸00.60^𝑛.
Find the smallest integer 𝑛, so that the design error, 𝐸𝑛 less than0.50%. Show the inequality and logarithm steps clearly.
To further improve the design, you introduced a boosting algorithm that reduces the error by 70% in the first two cycles and by 60% in each subsequent cycle, compute the new minimum 𝑛 to beat 0.50% error.
Each cycle takes 𝑡1 = 20minutes initially but becomes 10% quicker per cycle thereafter. If you stop at the 𝑛 cycles found in the first part without boosting, compute the total time 𝑇𝑛 =∑ 𝑡 𝑘 𝑛 𝑘=1
and its numerical value minutes.