# 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).