Chapter 4.5, Problem 80E

### Applied Calculus

7th Edition
Waner + 1 other
ISBN: 9781337291248

Chapter
Section

### Applied Calculus

7th Edition
Waner + 1 other
ISBN: 9781337291248
Textbook Problem

# Research and Development: Federal The total spent on research and development by the federal government in the United States during 2002–2012 can be approximated by S ( t ) = 3.1 ln t + 22  billion dollars   ( 2 ≤ t ≤ 12 ) , where t is the year since 2000. What was the total spent in 2005 ( t = 5 ) , and how fast was it increasing? [HINT: See Quick Examples 1 and 2.]

To determine

To calculate: The total spent by the federal government in 2005 that is, t=5 and also calculate the rate of its increase when the total spent on research and development by the federal government in United States during 20022012 is given by the function,

S(t)=3.1lnt+22

Where, t is the year since 2000 and S(t) is in billion dollars.

Explanation

Given information:

The function, S(t)=3.1lnt+22 represents the total spent on research and development by the federal government in United States during 2002âˆ’2012 where t is the year since 2000 and S(t) is in billion dollars.

Formula used:

1) Constant multiple rule of derivative of function f(x) is

f'(cx)=cf'(x)

Where, c is constant.

2) The derivative of natural logarithm of x is

ddxlnx=1x.

Calculation:

Consider the function, S(t)=3.1lnt+22

Where, t is the year since 2000 and S(t) is in billion dollars

The total spent in 2005 is calculated by substituting t=5 in the function S(t),

S(5)=3.1ln(5)+22=3.1â‹…1.6094+22=26.98914â‰ˆ27

Thus, the total spent by the federal government in 2005 is approximately \$27 billion dollars

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