Ratios, rates, proportions #8 - Questions and Answers | Georgia Southern University

University:
Georgia Southern University
Course:
MATH 2242 | Calculus II
Academic year:

2024
Views:

114

Pages:

2
Author:

Jarrett Brock

Question 1 The Otto-cycle engine in a Mercedes-Benz 230 has a compression ratio of 8.8. a) What is the ideal efficiency of the engine? Use 𝛾 = 1.40 γ=1.40. b) The engine in a Dodge Viper GT2 has a slightly higher compression ratio of 9.6. How much increase in the ideal efficiency results from this increase in the compression ratio? Answer Step 1: Ideal Efficiency Formula The ideal efficiency 𝑒 e for the Otto-cycle engine is given by: 𝑒 = 1 − 1 𝑟 𝛾 − 1 e=1− r γ−1 1 where 𝑟 r is the compression ratio and 𝛾 γ is the heat capacity ratio. Step 2: For 𝑟 = 8.8 r=8.8 and 𝛾 = 1.40 γ=1.40: 𝑒 = 1 − 1 ( 8.8 ) 1.40 − 1 = 1 − 1 ( 8.8 ) 0.4 e=1− (8.8) 1.40−1 1 =1− (8.8) 0.4 1 Calculating: 𝑒 = 1 − 1 1.75 ≈ 0.58 or 58 % . e=1− 1.75 1 ≈0.58 or 58%. Step 3: For 𝑟 = 9.6 r=9.6 and 𝛾 = 1.40 γ=1.40: 𝑒 = 1 − 1 ( 9.6 ) 0.4 . e=1− (9.6) 0.4 1 . Calculating: 𝑒 = 1 − 1 1.87 ≈ 0.594 or 59.4 % . e=1− 1.87 1 ≈0.594 or 59.4%. Step 4: Increase in Efficiency The increase in ideal efficiency is: Increase = 𝑒 𝑏 − 𝑒 𝑎 = 59.4 % − 58 % = 1.4 % . Increase=e b −e a =59.4%−58%=1.4%. Final Answer: The increase in ideal efficiency is 1.4%. Question 2 50 points decreased by 26%. Answer To find 26% of 50: 0.26 × 50 = 13. 0.26×50=13. Now subtract: 50 − 13 = 37. 50−13=37. Final Answer: The result is 37 points. Question 3 A regular pentagonal prism has 9-cm base edges. A larger, similar prism has 36-cm base edges. Compare the volumes. Answer 1 Scale Factor: The scale factor is: 36 9 = 4 : 1. 9 36 =4:1. Volume Ratio: The ratio of volumes of similar solids is the cube of the scale factor: ( 𝑎 𝑏 ) 3 = 4 3 = 64 : 1. ( b a ) 3 =4 3 =64:1. Final Answer: The volume of the larger prism is 64 times the volume of the smaller prism. Answer 2 The weight of similar objects is proportional to their volumes. Thus, the ratio of weights is also 64:1. Question 4 What is 60% of 200? Answer 1 𝑛 = 60 100 × 200 = 12000 100 = 120. n= 100 60 ×200= 100 12000 =120. Answer 2: 120 is 60% of 200. Question 5 What is 5% of 600? Answer 1 Convert 5% to a decimal: 5 % = 0.05. 5%=0.05. Multiply by 600: 0.05 × 600 = 30. 0.05×600=30. Answer 2 5 % 100 % × 600 = 30. 100% 5% ×600=30. Final Answer: 5% of 600 is 30.