In 1968, the U.S. minimum wage was $1.60 per hour. In 1976, the minimum wage was $2.30 per hour. Assume the minimum wage grows according to an exponential model w(t), where t represents the time in years after 1960.
Find the Formula for w(t)
My work so far:
n=y(not)
t=0
y=1.60
1.60=nb^0
n=1.6
t=8
y=2.30
2.30=1.60b^8
b=1.04640783
the final answer is suppose to be
y=1.113(1.04640783)^t
but i don't know how to find the 1.113
where did that come from?
I see an error in your first calculation
1960 ---> t = 0
1968 ---> t = 8, you had t = 0
so
1.6 = a(b)^8
2.3 = a(b)^16
divide the second equation by the first
1.4375 = b^8
b = 1.0464678
then in the first
1.6 = a(1.0464678)^8
1.6 = a(1.4375)
a = 1.6/1.4375
a = 1.11304
so w(t) = 1.11304(1.0464678)^t
Well, it seems like you're on the right track with finding the exponential growth rate, but you're missing one more piece of information to complete the formula.
To find the value of the constant, let's plug in the given values into the formula and solve for it.
We have:
1.6 = n * (1.04640783)^0
Since any number raised to the power of 0 is 1, this simplifies to:
1.6 = n * 1
Therefore, n = 1.6.
Now we can use this value of n to solve for the constant in the second equation:
2.3 = 1.6 * (1.04640783)^8
Divide both sides by 1.6:
2.3/1.6 = (1.04640783)^8
Taking the eighth root of both sides:
(2.3/1.6)^(1/8) = 1.04640783
This gives us the value of the constant as approximately 1.04640783.
Now, to find the missing 1.113, we can substitute the known values into the formula:
1.6 = 1.113 * (1.04640783)^0
Since any number raised to the power of 0 is 1, this simplifies to:
1.6 = 1.113 * 1
Therefore, 1.113 = 1.6.
So the complete formula for w(t), the U.S. minimum wage according to the exponential model, is:
w(t) = 1.113 * (1.04640783)^t
To find the value of 1.113 in the formula y = 1.113(1.04640783)^t, we can use one of the given data points.
In this case, the minimum wage in 1968 was $1.60 per hour. This corresponds to t = 8, as the time is measured in years after 1960. Plugging this information into the formula, we have:
1.6 = 1.113(1.04640783)^8
Now solve for 1.113:
1.6/1.04640783^8 = 1.113
After evaluating the calculation, we find that 1.6/1.04640783^8 is approximately 1.113. Therefore, the value of 1.113 can be derived from the given data.
To find the value of 1.113, we can use the given information that the minimum wage in 1968 was $1.60. Using this value, we can substitute it into the formula to solve for the unknown constant.
Let's write down the equation using the exponential model for the minimum wage:
w(t) = a * b^t
where:
w(t) represents the minimum wage at time t,
a represents the initial minimum wage at time t=0, and
b represents the growth factor.
We already know that in 1968 (t=0), the minimum wage was $1.60. Substituting this into the equation, we have:
1.60 = a * b^0
Since any value raised to the power of zero is 1, this simplifies to:
1.60 = a
So, the unknown constant "a" is equal to 1.60.
Substituting this value back into the equation, we have:
w(t) = 1.60 * b^t
Now we can continue with the information given for 1976 (t=8) where the minimum wage was $2.30:
2.30 = 1.60 * b^8
From your calculations, you found that b is approximately 1.04640783.
Plugging this value into the equation:
2.30 = 1.60 * (1.04640783)^8
Simplifying this expression will give you the final solution for w(t). Note that the actual value may have more decimal places, but rounding it to the nearest value is generally acceptable.