HW2

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University of California, Berkeley *

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11

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Mechanical Engineering

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Apr 3, 2024

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2

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CE 11 Homework #2 Due by 11:59 pm on Wednesday September 21 st . Make a pdf of your answers and upload to gradescope. Show your work, not just the final answer. Remember to tag your uploaded file so the reader knows which page(s) to look at to find your answers for each question. Question 1: Logistic Growth and World Population Assuming a carrying capacity for world population K = 15 billion, and a population of 7.75 billion as of 2020, estimate the year when population will reach 12 billion, assuming a logistic growth curve applies and with a current growth rate r 0 = 1.1% per year. What will the updated growth rate r f be in that future year when population reaches 12 billion? Question 2: Lithium Production in Australia Lithium is used to make batteries for products such as laptop computers, cell phones, cordless power tools, and electric vehicles. As of 2019, Australia was the largest producer of lithium worldwide, with P 0 = 51 000 t/yr. One metric ton (t) = 1000 kg, which is about 10% heavier than a US short ton which equals 2000 pounds. (a) Assuming exponential growth of production at a rate of 3.5% per year, how long would it take Australian lithium production to double? Also how many years will it take until remaining Australian lithium reserves of 2.7 million metric tons would be depleted? (b) Assuming a symmetric production curve with a peak production rate that is double the current rate (P 0 ), what will be the year when Australian lithium production peaks? What will the production rate be in 2050? Assume the total amount of lithium ever produced is 3 million metric tons, including both past and future years. In class we discussed worldwide lithium production, but the cited production and reserve numbers did not reflect a scale factor of 1000 that is shown on the x-axis of the world map/bar chart presented during lecture. The correct numbers for worldwide lithium production and reserves should be 85 000 t/yr and 15 million metric tons, respectively. Posted slides for Lecture #8 have been updated with these corrected values.
Question 3: Groundwater Pumping and Drawdown A pumping well at the origin is located at the interface between solid, impermeable granite to the west and a sand aquifer (K=50 m/day) to the east. Estimate the drawdown of the water table at the pumping well, given a pumping rate of 500 m 3 /day. The pumping well has a diameter of 0.2 m, and far away from the pumping well, the aquifer thickness is 30 m. Drawdown of the water table at a non-pumping well located 20 m to the east is observed to be 1 meter. See diagram of the site shown below. Question 4: Three-Well Technique for Finding Hydraulic Gradient Consider three monitoring wells, each located at the vertex of an equilateral triangle. The distance between any pair of wells is 300 m. The head (h) at each well, referenced to some common datum, is as follows: well 1, 100 m; well 2, 100.3 m; well 3, 100.3 m. Sketch the well field and find the magnitude and direction of the hydraulic gradient.
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