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

550

ccbxgf

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.