Question

farmers wishing to avoid the use of purchased 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, (square%) of the crop will be free of hybrid plants (round to the nearest integer as needed.)

Explanation:

Step1: Substitute $t = 5$ into $P(t)$

We are given $P(t)=(0.93)^t$. Substituting $t = 5$ gives $P(5)=(0.93)^5$.

Step2: Calculate $(0.93)^5$

$(0.93)^5=0.93\times0.93\times0.93\times0.93\times0.93\approx0.696$.

Step3: Convert to percentage

To convert to a percentage, we multiply by 100. So $0.696\times100 = 69.6\%\approx70\%$.

Step4: Find the derivative of $P(t)$

The derivative of $y = a^t$ with respect to $t$ is $y^\prime=a^t\ln(a)$. For $P(t)=(0.93)^t$, $P^\prime(t)=(0.93)^t\ln(0.93)$.

Step5: Evaluate $P^\prime(5)$

$P^\prime(5)=(0.93)^5\ln(0.93)$. We know $(0.93)^5\approx0.696$ and $\ln(0.93)\approx - 0.0725$. So $P^\prime(5)\approx0.696\times(- 0.0725)\approx - 0.0505$. The negative sign means the fraction of non - hybrid plants is decreasing at $t = 5$ years, and the magnitude $0.0505$ represents the rate of decrease of the fraction of non - hybrid plants per year at $t = 5$ years.

Step6: Solve for $t$ when half of the crop is hybrid

If half of the crop is hybrid, then the fraction of non - hybrid plants $P(t)=0.5$. So we set $(0.93)^t = 0.5$.

Step7: 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 $t\ln(0.93)=\ln(0.5)$.

Step8: Solve for $t$

$t=\frac{\ln(0.5)}{\ln(0.93)}\approx\frac{-0.6931}{-0.0725}\approx9.56$ years.

Answer:

a) $70$
b) $P^\prime(5)\approx - 0.0505$. It means the fraction of non - hybrid plants is decreasing at a rate of approximately $0.0505$ per year at $t = 5$ years.
c) Approximately $9.56$ years.