Question

In the 1990s the demand for personal computers in the home went up with household income. For a given community in the 1990s, the average number of computers in a home could be approximated by q = 0.3458 ln x − 3.045 10,000 ≤ x ≤ 125,000 where x is mean household income. A certain community had a mean income of $30,000, increasing at a rate of $1,000 per year.
a. How many computers per household were there? (Round your answer to four decimal places.) 0.5198 Correct: Your answer is correct. computers per household.
b. How fast was the number of computers in a home increasing? (Round your answer to four decimal places.) 0.0115 Correct: Your answer is correct. computers per household per year.

Answer 1

Answer:

a) 0.5198 computers per household

b) 0.01153 computers

Step-by-step explanation:

Given:

number of computers in a home,

q = 0.3458 ln x - 3.045 ;   10,000 ≤ x ≤ 125,000

here x is mean household income

mean income = $30,000

increasing rate, $\frac{dx}{dt}$ = $1,000

Now,

a) computers per household are

since,

mean income of  $30,000 lies in the range of 10,000 ≤ x ≤ 125,000

thus,

q = 0.3458 ln(30,000) - 3.045

or

q = 0.5198 computers per household

b) Rate of increase in computers i.e $\frac{dq}{dt}$

$\frac{dq}{dt}$ = $\frac{d(0.3458 ln x - 3.045)}{dt}$

or

$\frac{dq}{dt}=0.3458\times(\frac{1}{x})\frac{dx}{dt} - 0$

on substituting the values, we get

$\frac{dq}{dt}=0.3458\times(\frac{1}{30,000})\times1,000$

or

= 0.01153 computers