in fiscal year 2005, the federal government had revenues of $2.153 trillion and outlays of $2.472 trillion. for fiscal year 2006, revenues were $2.407 trillion and outlays were $.654 trillion
a) find an equation that relates the revenue R and the fiscal year x. Assume the relationship is linear
b) find an equation that relates the outlay D and the fiscal year x. Assume the relationship is linear
c) using the equations found in part a and b, determine the fiscal year in with the budget will be balanced. express all decimals round to three decimal places.
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a) To find an equation that relates the revenue R and the fiscal year x, we need to assume a linear relationship between the two variables. In this case, we can assume that the revenue increases linearly with each fiscal year.
Let's assign the fiscal year 2005 as x = 0. Then fiscal year 2006 will be x = 1, fiscal year 2007 will be x = 2, and so on.
Now, we need to find the equation of a line that passes through two points: (0, $2.153 trillion) and (1, $2.407 trillion).
Using the point-slope form of a linear equation, we have:
R - R₁ = m(x - x₁),
where R is the revenue, R₁ is the revenue for the year x₁, m is the slope of the line, x is the fiscal year, and x₁ is the fiscal year for R₁.
Let's substitute the values we have:
R - 2.153 = m(x - 0).
Since we have a second point, we can calculate the slope:
m = (R₂ - R₁) / (x₂ - x₁).
Substituting the values:
m = ($2.407 trillion - $2.153 trillion) / (1 - 0).
m = $0.254 trillion.
Now, we can substitute the slope and one of the points into the equation:
R - 2.153 = $0.254 trillion (x - 0).
Simplifying:
R - 2.153 = $0.254 trillion x.
The equation relating the revenue R and the fiscal year x is:
R = $0.254 trillion x + 2.153 trillion.
b) To find an equation that relates the outlay D and the fiscal year x, we can follow the same steps as in part a.
Using the given data points (0, $2.472 trillion) and (1, $0.654 trillion), we can calculate the slope:
m = (D₂ - D₁) / (x₂ - x₁)
m = ($0.654 trillion - $2.472 trillion) / (1 - 0)
m = -$1.818 trillion.
Substituting the slope and one of the points into the equation:
D - 2.472 = -$1.818 trillion (x - 0).
Simplifying:
D - 2.472 = -$1.818 trillion x.
The equation relating the outlay D and the fiscal year x is:
D = -$1.818 trillion x + 2.472 trillion.
c) To determine the fiscal year in which the budget will be balanced, we need to find the year where the revenue (R) equals the outlay (D).
Setting R equal to D in the two equations found in parts a and b:
$0.254 trillion x + $2.153 trillion = -$1.818 trillion x + $2.472 trillion.
Simplifying:
$2.153 trillion + $1.818 trillion x = $2.472 trillion - $0.254 trillion x.
Combining like terms:
$1.818 trillion x + $0.254 trillion x = $2.472 trillion - $2.153 trillion.
$2.072 trillion x = $0.319 trillion.
Dividing both sides by $2.072 trillion:
x = $0.319 trillion / $2.072 trillion.
x ≈ 0.154.
Rounding to three decimal places, the fiscal year in which the budget will be balanced is approximately 0.154.