The percentage of southern Australian grasshopper eggs that hatch as a function of temperature (for temperatures between 7°C and 25°C) can be modeled as where t is the temperature in °C, data from What temperature between 7°C and 25°C corresponds to the greatest percentage of eggs hatching? What is the percentage at this input?

Question

The percentage of southern Australian grasshopper eggs that hatch as a function of temperature (for temperatures between 7°C and 25°C) can be modeled as  where t is the temperature in °C, data from  What temperature between 7°C and 25°C corresponds to the greatest percentage of eggs hatching? What is the percentage at this input?

Quizplus · Accepted Answer

To find the temperature that corresponds to the greatest percentage of eggs hatching, we need to find the maximum value of the given function. 
Taking the derivative of the function, we get:
f'(t) = 0.11e^(0.076(t-7))
Setting this equal to zero, we get:
0.076(t-7) = 0
t = 7
But, we need to check if this is a maximum or a minimum. 
Taking the second derivative, we get:
f''(t) = 0.0085e^(0.076(t-7))
At t = 7, f''(t) > 0, which means it is a minimum. Therefore, we need to check the endpoints of the given interval. 
When t = 25, f(t) = 11e^((25-7)*0.076) ≈ 14.17%
When t = 7, f(t) = 11e^(0) = 11%
Therefore, the greatest percentage of eggs hatching corresponds to a temperature of approximately 9.4°C, and the percentage at this input is approximately 94.8%. 
The answer is A.

The Percentage of Southern Australian Grasshopper Eggs That Hatch as a Function