Question

farmers wishing to avoid the use of patented seeds are increasingly concerned about inadvertently growing hybrid plants as a result of pollen - drifting from nearby farms. assuming that these farmers raise their own seeds, the fractional portion of their crop that remains free of hybrid plants t years later can be approximated by p(t)=(0.93)^t.
a) using this model, predict the fractional portion of the crop that will be free of hybrid plants 5 yr after a neighboring farm begins to use purchased seeds.
b) find p(5) and explain its meaning.
c) when will half of the crop be hybrid plants?
a) after 5 yr, % of the crop will be free of hybrid plants.
(round to the nearest integer as needed.)
b) p(5)= % of the crop per year.
(round to one decimal place as needed.)
which of the following is the correct interpretation of p(5)?
a. in 5 years, the percentage of the crop that is hybrid is changing by p(5).
b. in 5 years, the percentage of the crop that remains free of hybrid plants is changing by p(5).
c. in 5 years, the percentage of the crop that is hybrid is p(5).
d. in 5 years, the percentage of the crop that remains free of hybrid plants is p(5).
c) half of the crop will be hybrid plants in years.
(round to two decimal places as needed.)

Explanation:

Step1: Identify the function

Let \(P(t)\) be the fractional - portion of the crop that remains free of hybrid plants \(t\) years after a neighboring farm begins to use purchased seeds. Assume \(P(t)=(0.93)^t\).

Step2: Predict the fractional portion at \(t = 5\)

Substitute \(t = 5\) into the function \(P(t)\).
\[P(5)=(0.93)^5=0.93\times0.93\times0.93\times0.93\times0.93\approx0.696\]

Step3: Find \(P^{\prime}(t)\)

First, if \(y = a^x\), then \(y^\prime=a^x\ln a\). For \(P(t)=(0.93)^t\), \(P^{\prime}(t)=(0.93)^t\ln(0.93)\). \(P^{\prime}(5)=(0.93)^5\ln(0.93)\approx0.696\times(- 0.0725)\approx - 0.0505\). The negative sign indicates that the percentage of non - hybrid plants is decreasing. \(P^{\prime}(5)\) represents the rate of change of the percentage of the crop that remains free of hybrid plants at \(t = 5\) years.

Step4: Find when half of the crop is hybrid

If half of the crop is hybrid, then the fraction of non - hybrid plants is \(P(t)=0.5\).
Set \((0.93)^t = 0.5\).
Take the natural logarithm of both sides: \(\ln((0.93)^t)=\ln(0.5)\).
Using the property \(\ln(a^b)=b\ln(a)\), we get \(t\ln(0.93)=\ln(0.5)\).
Then \(t=\frac{\ln(0.5)}{\ln(0.93)}\approx\frac{-0.693}{-0.0725}\approx9.56\).

Answer:

a) Approximately \(0.696\)
b) \(P^{\prime}(t)=(0.93)^t\ln(0.93)\), \(P^{\prime}(5)\approx - 0.0505\), and the correct interpretation is: In 5 years, the percentage of the crop that remains free of hybrid plants is changing by \(P^{\prime}(5)\)
c) Approximately \(9.56\) years