because of a slump in the economy, a company finds thatr its annual revenue have dropped from $724,000 in 2006 to $632,000 in 2008. if the revenue is following an exponential ppattern of decline, what is the expected revenue for 2009? (Let t=0 represent 2006.)
revenue=last year*e^kt
solve for k (given t=2, and the figures above.
k2=ln(632/724)
I get k to be-.0680
Now solve for 2009
revenue=reventue2006 * e^(-.0680t) where t is three years.
To find the expected revenue for 2009, we can use the formula for exponential growth or decline:
P(t) = P0 * (1 - r)^t
Where:
P(t) represents the revenue at time t,
P0 represents the initial revenue,
r represents the growth/decline rate, and
t represents the number of years.
In this case, we want to find the revenue for 2009, which would be t = 2009 - 2006 = 3.
Given that the revenue in 2006 (t = 0) is $724,000 and the revenue in 2008 (t = 2) is $632,000, we can substitute these values into the equation:
$632,000 = $724,000 * (1 - r)^2
Divide both sides of the equation by $724,000:
$632,000 / $724,000 = (1 - r)^2
Now, take the square root of both sides to isolate (1 - r):
√(632,000 / 724,000) = 1 - r
Calculate the value inside the square root:
√(0.8729) = 1 - r
Now subtract the square root value from 1 to find r:
1 - √(0.8729) = r
r ≈ 0.169
Now that we have the value of r, we can calculate the expected revenue for 2009 (t = 3):
P(3) = $724,000 * (1 - 0.169)^3
P(3) = $724,000 * (0.831)^3
P(3) ≈ $724,000 * 0.573
P(3) ≈ $415,452
Therefore, the expected revenue for 2009 is approximately $415,452.
To find the expected revenue for 2009, we need to determine the exponential pattern of decline and extrapolate it to the next year.
The exponential pattern of decline can be represented by the formula:
R = A * e^(kt)
Where:
R = Revenue at a given time t
A = Initial revenue
k = Growth/decay rate
t = Time
In this case, the initial revenue in 2006 (t=0) is $724,000, and the revenue in 2008 (t=2) is $632,000. So we can set up two equations to find the values of A and k:
Equation 1: 632,000 = 724,000 * e^(k*2) (substituting values of t=2 and R=632,000 in the formula)
Equation 2: 724,000 = 724,000 * e^(k*0) (substituting values of t=0 and R=724,000 in the formula)
Now, let's solve these equations simultaneously:
Equation 2 simplifies to: 1 = e^0
So, k * 0 = 0
Equation 1 simplifies to: 632,000/724,000 = e^(k*2)
By dividing both sides by 724,000, we get: 0.8724 = e^(2k)
Next, we take the natural logarithm (ln) of both sides to solve for k:
ln(0.8724) = ln(e^(2k))
ln(0.8724) = 2k * ln(e)
ln(0.8724) = 2k
Now, we can solve for k by dividing both sides by 2:
k = ln(0.8724) / 2
Using a calculator, we find that k is approximately -0.064.
Now that we have the value of k (-0.064), we can find the expected revenue for 2009 (t=3):
R(3) = A * e^(k*3)
Substituting the given values:
R(3) = 724,000 * e^(-0.064*3)
Using a calculator, we find that R(3) is approximately $576,264.
Therefore, the expected revenue for 2009 is approximately $576,264.