The projected public spending by the British government for the years 2010-2011 (t = 0) through 2014-2015 (t = 4) is 692 billion pounds/year. The projected public revenue for the same period is f(t) = 292.6t + 134.4 / t + 6.9 + 500 billion pounds (0 ge t ge 4). Find the projected deficit for the period in question. (Round your answer to one decimal place.) billion pounds
To find the projected deficit for the period in question, we need to calculate the difference between the projected public spending and the projected public revenue.
Given:
Projected public spending for the years 2010-2011 (t=0) through 2014-2015 (t=4) is 692 billion pounds/year.
Projected public revenue for the same period is given by the equation:
f(t) = 292.6t + 134.4 / t + 6.9 + 500 billion pounds (0 ≤ t ≤ 4).
To find the projected deficit, we need to substitute the values of t from 0 to 4 into the equation for projected public spending and projected public revenue, and then calculate the difference.
Let's calculate for each year separately and then sum the results.
For t = 0:
Projected public spending = 692 billion pounds/year
Projected public revenue = f(0) = 292.6(0) + 134.4 / 0 + 6.9 + 500 = 500 billion pounds/year
Deficit for t = 0: 500 - 692 = -192 billion pounds/year (negative means deficit)
For t = 1:
Projected public spending = 692 billion pounds/year
Projected public revenue = f(1) = 292.6(1) + 134.4 / 1 + 6.9 + 500 = 500 + 292.6 + 134.4/7.9 = 500 + 37.1 + 17 = 554.1
Deficit for t = 1: 554.1 - 692 = -137.9 billion pounds/year
Similarly, we can calculate for t = 2, t = 3, and t = 4.
For t = 2:
Projected public spending = 692 billion pounds/year
Projected public revenue = f(2) = 292.6(2) + 134.4 / 2 + 6.9 + 500 = 500 + 585.2 + 134.4/8.9 = 500 + 65.6 + 15.1 = 580.7
Deficit for t = 2: 580.7 - 692 = -111.3 billion pounds/year
For t = 3:
Projected public spending = 692 billion pounds/year
Projected public revenue = f(3) = 292.6(3) + 134.4 / 3 + 6.9 + 500 = 500 + 877.8 + 134.4/9.9 = 500 + 88.7 + 13.6 = 602.3
Deficit for t = 3: 602.3 - 692 = -89.7 billion pounds/year
For t = 4:
Projected public spending = 692 billion pounds/year
Projected public revenue = f(4) = 292.6(4) + 134.4 / 4 + 6.9 + 500 = 500 + 1170.4 + 134.4/10.9 = 500 + 107.3 + 12.3 = 619.6
Deficit for t = 4: 619.6 - 692 = -72.4 billion pounds/year
To find the total projected deficit for the period in question, we sum up the deficits for each year:
Total projected deficit = (-192) + (-137.9) + (-111.3) + (-89.7) + (-72.4)
Total projected deficit = -602.3 billion pounds/year
Therefore, the projected deficit for the period in question is -602.3 billion pounds/year (rounded to one decimal place).