A farmer grows a particular plant that has a gene that can be manipulated to control the age t at which the plant matures. The number of seeds S(t) produced by a plant maturing at age t is S(t)=−0.3t2+30t+0.2 seeds per mature plant A farmer asks the geneticists to genetically engineer a plant line that accounts for the fact that on his farm, only P(t)=90000t+100, plants mature to age t. What age of maturity should the geneticist select for the plants to maximize the seed production of the farmer's crop?a. Between 130 and 140 days b. Between 120 and 130 days

The correct option is;

Between 40 and 50 days

Step-by-step explanation:

The number of seeds that are produced by a plant maturing at age t, S(t), is given as follows;

S(t) = -0.3·t² + 30·t + 0.2

The proportion of plants maturing at age (t) in the plants to be engineered by the geneticist P(t) = 90000/(t + 100)

The number of seeds produced by the plants  = S(t) × P(t) = (-0.3·t² + 30·t + 0.2)×(90000/(t + 100))

To find the maximum number of seeds, we differentiate using an online tool, and equate to zero to get;

d((-0.3·t² + 30·t + 0.2)×(90000/(t + 100)))/dt = (-27000·t² - 5400000·t + 269982000)/(t + 100)² = 0

(-27000·t² - 5400000·t + 269982000)/(t + 100)² = 27000(t - 41.419)(t + 241.419)/(t + 100)² = 0

t = 41.419 or t = -241.419

Therefore, in order to maximize the production of seed of the crops of the farmer, the geneticist should select between 40 and 50 days.

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step-by-step explanation: