Question

Question 9 A restaurant purchased a new-brand coffee machine for $1 550 and it depreciates by 8 cents for every cup of coffee it makes.

a. Write down the recurrence relation which models the depreciation of the coffee machine?

b. Write down the rule for the value of the machine after making n cups of coffee?

c. Find the value of the coffee machine after 275 cups of coffee are made.

d. On average, the coffee machine is used to make 625 cups of coffee each week. What is the value of the machine after 6 months?

Answer 1

9. Given,
Initial cost of coffee machine, Po = $1550
depreciatino rate, r = 8c per cup
now, we will consider two cases as the above statement is ambiguous

case 1 : rate r = 8c per cup = $0.08 per cup
case 2 : rate r = 8% per cup

a.
Recurrence equation for case 1 will be, after n cups of coffee is
Pn = Pn-1 - 0.08
for case 2, afer n cups recurrence relation is
Pn = Pn-1(1 - 0.08)

  
b. hence after n cups for coffee, the rule for price for case 1 is
Pn = Po - nr = Po - 0.08n

for case 2
dP = -0.08*P*dn
hence
ln(Pn/Po) = -0.08n
Pn = Po*e^(-0.08n)
  
c. n = 275
from case 1: P = 1550 - 0.08*275 = $ 1528
from case 2 : P = 1550*e^(-0.08*275) = $4.323 *10^-7 [ this seems unrealistic, this means the second case is not the one]
  
d. So we consider case 1
n = 625 / 7 = 89.285714285714 cups per day
in 6 months, n = 365*89.285714285/2 = 16294.6428 cups
P = 1550 - 16294.64285714(0.08) = $ 246.42857