Question

farmers wishing to avoid the cost of purchased seeds are increasingly concerned about accidentally 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)=(0.98)^t).
a) using this model, predict the fractional portion of the crop that will be free of hybrid plants 15 yr after a neighboring farm begins to use purchased seeds.
b) find (p(15)) and explain its meaning.
c) when will half of the crop be hybrid plants?
a) after 15 yr, (%) of the crop will be free of hybrid plants. (round to the nearest whole number as needed.)
b) (p(15)=) (%) of the crop per year. (round to one - decimal place as needed.)
which of the following is the correct interpretation of (p(15))?
a. in 15 years, the percentage of the crop that remains free of hybrid plants is changing by (p(15)).
b. in 15 years, the percentage of the crop that is hybrid is changing by (p(15)).
c. in 15 years, the percentage of the crop that is hybrid is (p(15)).
d. in 15 years, the percentage of the crop that remains free of hybrid plants is (p(15)).
c) half of the crop will be hybrid plants in (square) years. (round to two decimal places as needed.)

Explanation:

Step1: Predict fraction for part a

Substitute \(t = 15\) into \(F(t)=(0.98)^t\).
\[F(15)=(0.98)^{15}\]
\[F(15)\approx0.74\]

Step2: Find derivative for part b

First, find the derivative of \(F(t)=(0.98)^t\). Using the formula \((a^t)^\prime=a^t\ln(a)\), we have \(F^\prime(t)=(0.98)^t\ln(0.98)\). Then substitute \(t = 15\) into \(F^\prime(t)\).
\[F^\prime(15)=(0.98)^{15}\ln(0.98)\]
\[F^\prime(15)\approx0.74\times(- 0.0202)\approx - 0.015\]
The meaning of \(F^\prime(15)\) is that in 15 years, the percentage of the crop that remains free of hybrid - plants is changing by \(F^\prime(15)\) (a decrease since it's negative).

Step3: Solve for part c

We want to find \(t\) when the fraction of non - hybrid plants is \(0.5\). So we set \(F(t)=(0.98)^t = 0.5\).
Take the natural logarithm of both sides: \(\ln((0.98)^t)=\ln(0.5)\).
Using the property \(\ln(a^b)=b\ln(a)\), we get \(t\ln(0.98)=\ln(0.5)\).
\[t=\frac{\ln(0.5)}{\ln(0.98)}\]
\[t\approx34.3\]

Answer:

a) \(0.74\)
b) \(F^\prime(15)\approx - 0.015\); In 15 years, the percentage of the crop that remains free of hybrid plants is changing by \(F^\prime(15)\)
c) \(34.3\)