Business Calculus

Question

A glacier covered about 177 acres in 2004 and was shrinking at a rate of about 3.1% per year.

(a) Write a formula for the size, , of the glacier, in acres, as a function of years  since 2004. ______

(b) Use the model to predict the size of the glacier in the year 2014.Size:___________  acres

(c) According to the model, how many acres of ice were lost from the glacier between 2004 and 2009?Acres lost: ____

Answer 1

This problem is a way to work with negative growth rates / time value of money / compounding interest. Its just that instead of money changing, its the acres of the glacier. We can just use a negative rate of change in the normal formula.

The relevant formula is:

Future value = present value * (1+ rate of change)^periods

a) The formula would be area in 2020 = 177* (1-3.1/100)¹⁶ = 106.94 acres in 2020.

b) from 2004 to 2014 is 10 years so just use n = 10 in the above.

area in 2014 = 177 * (1-0.031)¹⁰ = 129.18

c) area in 2004 - area in 2009 = 177 - 177 * (1-0.031)^{5 =} 177 - 151.21 = 25.79 acres lost.

Another way to see this is if you have a business calculator, use the present value as 177, the interest rate as -3.1%, the term as 16 in a), 10 in b) and 5 in c) and press the future value button.

Specifically,for a) n= 16, i = -3.16%, pv=177, pmt = 0 and fv = ______ push the button and read it off.

Then for b), just change n to 10 and push FV

then for c) just change n to 5 and push FV and add 177.

Note that if you do this, the future value will be negative since it is assuming cashflows, (negative for money going out, positive for money coming in) of course, area is only positive.

Answer 2

Part A

F(t) = 177 (0.969^t)

Part b

Use the calculator to find F(10)

Part c

Use the calculator to find f(5)-177

Answer 3

Well 100 % - 3.1 % = 96.9 % = 0.969

So for a . if the area is A (t) acres t years after 2004, A(t) = 177 (0.969)^t

For b, find A(10)

For c, find A(0) - A(5)