Question

f(x) = -x³ + 6x² + x x₁ = 1, x₂ = 6

1. applied research the amounts (in billions of dollars) the u.s. federal government spent

on applied research from 2013 through 2018 can be approximated by the model
y = 0.0729t² - 0.526t + 24.34
where t represents the year, with t = 13 corresponding to 2013. (source: national center for science
and engineering statistics)

a. use a graphing utility to graph the model

b. find the average rate of change of the model from 2013 to 2018. interpret your answer in
the context of the problem

Explanation:

Part b:

Step1: Identify t values

For 2013, \( t = 13 \); for 2018, \( t = 18 \).

Step2: Find y at t=13 and t=18

At \( t = 13 \):
\( y(13)=0.0729(13)^2 - 0.526(13)+24.34 \)
\( = 0.0729(169)-6.838 + 24.34 \)
\( = 12.3201 - 6.838 + 24.34 = 29.8221 \)

At \( t = 18 \):
\( y(18)=0.0729(18)^2 - 0.526(18)+24.34 \)
\( = 0.0729(324)-9.468 + 24.34 \)
\( = 23.6196 - 9.468 + 24.34 = 38.4916 \)

Step3: Calculate average rate of change

Average rate of change \(=\frac{y(18)-y(13)}{18 - 13}=\frac{38.4916 - 29.8221}{5}=\frac{8.6695}{5}=1.7339\)

Answer:

The average rate of change is approximately \( 1.73 \) billion dollars per year. This means the U.S. federal government’s spending on applied research increased by about \( 1.73 \) billion dollars per year from 2013 to 2018.