Barnaby’s grandfather is always complaining that back when he was a teenager, he used to be able to buy his girlfriend dinner for only.
A. If that same dinner that Barnaby’s grandfather purchased for
sixty years ago now costs
, and the price has increased exponentially, write an equation that will give you the costs at different times.
B.How much would you expect the same dinner to cost in 60 years?
(current price) = (old price) * [1 + (annual increase)]^(years)
B. (future price) = (current price) * [1 + (annual increase)]^(years)
annual increase is the percentage , entered as a decimal
how about some numbers?
Knowing P1 and P2, you need to find r such that
P1(1+r)^60 = P2
B: = $425.04
To solve this problem, we need to use the concept of exponential growth and create an equation. Exponential growth occurs when a quantity increases at a constant percentage rate over a given period of time.
Let's assume the initial cost of the dinner that Barnaby's grandfather used to purchase 60 years ago is represented by "P0" (in dollars). We know that the price has increased exponentially over time. Therefore, the equation representing this exponential growth can be written as:
P = P0 * (1 + r)^t
Where:
P is the cost of the dinner at a specific time (in dollars),
P0 is the initial cost of the dinner (in dollars),
r is the rate of increase (expressed as a decimal or percentage), and
t is the time (in years).
Now, using the information given, we have P0 = 3 and P = 9. We also know that the time, t, is 60 years. However, the rate of increase, r, is not provided. Therefore, we need to solve for r.
To find r, we can rearrange the formula:
(1 + r)^t = P / P0
Substituting the given values:
(1 + r)^60 = 9 / 3
Simplifying further:
(1 + r)^60 = 3
To solve for r, we can take the 60th root of both sides of the equation:
1 + r = √3
Now subtracting 1 from both sides:
r = √3 - 1
Calculating the value of r:
r ≈ 0.732
Therefore, the equation representing the costs at different times is:
P = 3 * (1 + 0.732)^t
Now, to find the cost of the same dinner in 60 years, we substitute t = 60 into the equation:
P = 3 * (1 + 0.732)^60
Using a calculator, we find:
P ≈ 164.41
Therefore, we would expect the same dinner to cost approximately 164.41 dollars in 60 years.