The cost of a Hershey bar in 1964 was $0.05. It had risen to $1.91 in 2020, exhibiting exponential growth during this period
a. Find the exponential function that fits this data, describing the price of the Hershey bar since 1964.
b. If this rate of exponential growth continues, to the nearest cent how much will the Hershey bar cost in 2030?
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To find the exponential function that fits this data, we can use the formula for exponential growth:
P(t) = P₀ * e^(kt)
where P(t) is the price at time t, P₀ is the initial price, e is Euler's number (approximately 2.718), k is the growth rate, and t is the time.
We know that the price in 1964 (P₀) is $0.05, so P₀ = 0.05. We can use the price in 2020 (P) to find the growth rate (k).
P = 1.91 (price in 2020)
0.05 = 1.91 * e^(k * 2020)
Dividing both sides by 1.91:
(e^(k * 2020)) ≈ 0.05/1.91
Taking the natural logarithm of both sides to isolate the exponent:
ln(e^(k * 2020)) ≈ ln(0.05/1.91)
Using the property of logarithms, we can bring down the exponent:
k * 2020 ≈ ln(0.05/1.91)
Dividing both sides by 2020 to solve for k:
k ≈ ln(0.05/1.91) / 2020
Using a calculator:
k ≈ -0.034
Now we have the growth rate. We can substitute this value of k into the formula to find the exponential function:
P(t) = 0.05 * e^(-0.034t)
To find the cost in 2030 (t = 2030-1964 = 66), we substitute t = 66 into the function:
P(66) = 0.05 * e^(-0.034 * 66)
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