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)=(0.98)^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, 74% of the crop will be free of hybrid plants. (round to the nearest integer as needed.)
b) p(15)= - 1.5% 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 years. (round to two decimal places as needed.)

Explanation:

Step1: Find the fractional portion at t = 15 for part a

Given $P(t)=(0.88)^t$, substitute $t = 15$.
$P(15)=(0.88)^{15}\approx0.1469$. As a percentage, it is $0.1469\times100\approx15\%$. So the fractional portion free of hybrid plants is approximately $1 - 0.15= 0.85$ or $85\%$ (contradicting the given answer in the image which seems to have an error in calculation in the image).

Step2: Differentiate $P(t)$ for part b

The derivative of $y = a^t$ with $a = 0.88$ is $y'=a^t\ln(a)$. So $P'(t)=(0.88)^t\ln(0.88)$. Substitute $t = 15$.
$P'(15)=(0.88)^{15}\ln(0.88)\approx0.1469\times(- 0.1278)\approx - 0.0187$. As a percentage per year, it is $-1.9\%$ (rounded to one - decimal place). The correct interpretation of $P'(15)$ is that in 15 years, the percentage of the crop that remains free of hybrid plants is changing by $P'(15)$.

Step3: Solve for $t$ when half of the crop is hybrid for part c

If half of the crop is hybrid, then the fractional portion free of hybrid plants is $P(t)=0.5$. So we set $(0.88)^t = 0.5$.
Take the natural logarithm of both sides: $\ln((0.88)^t)=\ln(0.5)$. Using the property $\ln(a^b)=b\ln(a)$, we get $t\ln(0.88)=\ln(0.5)$. Then $t=\frac{\ln(0.5)}{\ln(0.88)}\approx5.42$ years.

Answer:

a) $85\%$
b) $P'(15)\approx - 1.9\%$ of the crop per year; Interpretation: In 15 years, the percentage of the crop that remains free of hybrid plants is changing by $P'(15)$
c) $5.42$ years