六、請回答下列問題：(4 題，共 15 分)
(一)質傳通量的式子為：$Flux = -D_{AB} \frac{dc}{dz}$，請問與之相似的動量及熱量之通量式子為何？說明符號之意義及彼此相似性。(5 分)

(1) $Schmidt\ number = \frac{momentum\ diffusivity}{mass\ diffusivity}$(2 分)

(2) $Lewis\ number = \frac{thermal\ diffusivity}{mass\ diffusivity}$(2 分)

(三)在熱傳中與 Sc 對等之無因次數字為何？其定義又為何？(2 分)

(四)質傳中之 Sherwood number 與熱傳中之 Nusselt number 相似，其定義分別為何？(4 分)

1. Suppose the market supply and demand for milk can be expressed as:$$Q^S = 3P$$$$Q^D = 100 - 2P,$$If the government imposes a $5 unit tax to be collected from sellers, how much tax revenue is collected? What fraction is paid by sellers? (10%)

a. Paul's utility function is \( U = 2x + 3y \). (5%)

b. Paul's utility function is \( U = \min \{x, 3y\} \). (5%)

3. James has a monopoly on a board game called "Fly High" and faces a demand curve given by \( Q_T = 100 - P \), where \( Q_T \) equals the combined total number of games produced per hour in the company's two factories (\( Q_T = q_1 + q_2 \)). If factory 1 has a marginal cost function given by \( MC_1 = q_1 - 5 \) and factory 2 has a marginal cost function given by \( MC_2 = 0.5q_2 - 5 \), how much total output will James choose to produce and how will he divide this output between its two factories in order to maximize profits? (10%)

4. XYZ company has a Cobb-Douglas production function, \( Q = L^{0.5}K^{0.5} \). Use \( w \) and \( r \) to denote the price of the two inputs, \( L \) and \( K \), respectively. Given a required output level \( \bar{Q} \), compute XYZ's conditional factor demand, \( L(w, r; \bar{Q}) \) and \( K(w, r; \bar{Q}) \). (10%)

a. What is the optimal level of mosquito control? (5%)

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