Summary TUTORIAL 8B MEMO

STK120 – 162 TUT 8B MEMO

Question 8 (TUT8B)
The coffee shop on campus offers breakfast on a daily basis. The coffee shop has a fixed cost of R800
per day and the breakfast ingredients amounts to R10 per plate of food. From experience the owner
knows that he sells 100 10𝑥) plates of breakfast per day when he charges R 𝑥 per plate.

a. The cost function for the coffee shop per day is:
𝐶𝑜𝑠𝑡 10 100 10𝑥 800 1000 100𝑥 800 1800 100𝑥
b. The number of plates of breakfast that maximises the coffee shop’s income is:
𝐼𝑛𝑐𝑜𝑚𝑒 𝑥 100 10𝑥 100𝑥 10𝑥 ∴ 100 20𝑥 ∴ 100 20𝑥 0 ∴ 𝑥 5

𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑙𝑎𝑡𝑒𝑠 100 10 5 50
Question 9 (TUT 8B)

Calculate and for the following functions:

a. 𝑧 8𝑥 2𝑥𝑦 3𝑥 𝑦 1
𝜕𝑧
24𝑥 2𝑦 6𝑥
𝜕𝑥
𝜕𝑧
2𝑥 2𝑦
𝜕𝑦
b. 𝑧 4𝑦 𝑒
𝜕𝑧
6𝑥𝑒
𝜕𝑥
𝜕𝑧
12𝑦 2𝑒
𝜕𝑦
c. 𝑧 6𝑥 𝑙𝑛 𝑥 5𝑦
𝜕𝑧 4𝑥 24𝑥
6𝑥 𝑙𝑛 𝑥 5𝑦 . 18𝑥 𝑙𝑛 𝑥 5𝑦 . 18𝑥
𝜕𝑥 𝑥 5𝑦 𝑥 5𝑦
𝜕𝑧 5 30𝑥
6𝑥 𝑙𝑛 𝑥 5𝑦 . 0
𝜕𝑦 𝑥 5𝑦 𝑥 5𝑦

Question 10 (TUT 8B)
A Bakery bakes special bread on a daily basis. The bakery bakes x white breads and y brown breads
daily. The price of a white bread is 15𝑥 rand and the price of a brown bread is 12𝑦 rand.
It is known that the profit to bake x white and y brown breads are:
𝑥 𝑦
𝑃 𝑥, 𝑦 15𝑥 12𝑦
12 20

a. 15 15
𝜕𝑃 2𝑦 𝑦
12 12
𝜕𝑦 20 10
;
b. The interpretation of the marginal profit function 11, is:
Profit increases with R11 if an extra (24 to 25) white bread is baked, when the number of brown
breads stays constant at 30.
;
c. The interpretation of the marginal profit function 9, is:
Profit increases with R9 if an extra (30 to31) brown bread is baked, when the number of white
breads stays constant at 24.