i don't understand this word problem.
Barnany's godfather is alway complaining that back when he was a teenager, he used to be able to buy his girlfriend dinner for only $1.50
a) If that same dinner that Barnaby's grandfather purchased for $1.50 sixty years ago now costs $25.25, and the price has increased exponentielly, write and equation that will give you the costs at different times.
b) How much would you expect the same dinner to cost in sixty years?
One last smart remark. The "same dinner" in sixty more years is going to be a tough chew, even if it was flash frozen. I wouldn't pay much for it.
a) To write an equation that gives the costs at different times, we can use the concept of exponential growth. Let's assume the cost of the dinner increases exponentially with time.
Let's denote the initial cost of the dinner as "C₀" and the number of years since then as "t". We are given that the dinner's cost 60 years ago, C₀, was $1.50, and the current cost, C, is $25.25.
The equation for exponential growth can be written as:
C = C₀ * (1 + r)^t
Where:
C is the current cost of the dinner
C₀ is the initial cost of the dinner
r is the rate of growth (expressed as a decimal)
t is the number of years
In this case, we need to find the rate of growth, r. We can use the given information to calculate it.
To find the rate of growth, we can use the formula:
r = (C / C₀)^(1/t) - 1
Substituting the given values:
C₀ = $1.50
C = $25.25
t = 60
r = ($25.25 / $1.50)^(1/60) - 1
Now, we can write the equation for the costs at different times using the calculated value of r:
C = $1.50 * (1 + r)^t
b) To find the cost of the dinner in sixty years, we can substitute the value of t = 60 years into the equation we derived in part a:
C = $1.50 * (1 + r)^60
Simply compute this expression to find the expected cost of the dinner in sixty years.
To understand this word problem, let's break it down step by step.
a) The problem states that the dinner's price has increased exponentially over the span of sixty years. To write an equation that represents this exponential growth, we need to find the growth rate and the initial cost of the dinner.
Let's denote the initial cost (back when Barnaby's grandfather purchased it) as C₀, and the current cost as Cₙ after n years. The problem tells us that C₀ = $1.50 and Cₙ = $25.25 (after sixty years).
Exponential growth is generally represented as: Cₙ = C₀ * (1 + r)^n, where r is the growth rate and n is the number of years.
Substituting the given values, we have $25.25 = $1.50 * (1 + r)^60.
b) To find out how much the same dinner would cost in sixty years, we can rearrange the equation from part a to solve for Cₙ.
Dividing both sides by $1.50 and then taking the 60th root of both sides, we get:
(25.25 / 1.50)^(1/60) = (1 + r).
Now, substitute this value into the exponential growth equation and solve for Cₙ:
Cₙ = $1.50 * (1 + (25.25 / 1.50)^(1/60))^60.
Calculating this expression will give you the expected cost of the dinner in sixty years.
Note: It's important to remember that this model assumes the exponential growth rate remains constant over time, which might not always be the case in real-world scenarios.
Cost(t)=originalcost*ek*t
Given at t=0,Cost=1.50 solve for original cost.
at t=60, Cost =25.25,solve for k
Finally, solve it when t=120