Question

Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s. For example, che value of a particular rocking chair is well approximated byV = 60(1.25)^twhere V is in dollars and t is the number of years since 1975. Find the rate, in dollars per year, at which the price is increasing.Rate= ____ dollars/yr

Answer 1

Step 1:

In this question, we are given the following:

Certain pieces of antique furniture increased very rapidly in price in the 1970s and 1980s.

For example, the value of a particular rocking chair is well approximated by:

`V=60(1.25)^t`

where V is in dollars and t is the number of years since 1975.

Find the rate, in dollars per year, at which the price is increasing.

Step 2:

Next,

`\begin{gathered} \text{Given,} \\ V=\text{ }60(1.25)^t \\ \text{Calculating the rate, } \\ we\text{ ne}ed\text{ to differentiate V with respect to t, we have that:} \\ (dV)/(\differentialDt t)=(1.25)^t\ln (1.25)\text{ 60} \end{gathered}`

Now, we need to evaluate:

`\begin{gathered} \ln (1.25)60\text{ = 13. 38861308 }\approx\text{ 13. 3886} \\ \end{gathered}`

Hence, the rate, in dollars per year since 1975, at which the price is increasing is:

`(dV)/(\differentialDt t)=13.3886(1.25)^t`