The codling moth is an insect that causes serious damage to apples. Adult codling moths emerge from their cocoons in the spring. They soon mate, and the female lays as many as 130 tiny eggs on the leaves of apple trees. After the codling moth larva, also known as the common apple worm, hatches from its egg, it goes looking for an apple. The time between hatching and finding an apple is called the searching period. Once a codling moth finds an apple, it squirms inside and eats the fruit and seeds of the apple, thereby ruining it. The codling moth backs out of the apple and crawls under the bark of the tree or into the soil where it forms a cocoon. Observations regarding the behavior of codling moths indicate that the length of the searching period, S(T), and the percentage of larvae that survive the searching period, N(T), depend on the air temperature, T. Methods of data analysis (polynomial regression) applied to data recorded from observations suggest that if T is measured in degrees Celsius with 20 ≤ T ≤ 30, then S(T) and N(T) may be modeled* by S(T) = (-0.03T²+1.6T-13.65)^-1 days and N(T) = -0.85T²+ 45.4T - 547.

Part 1) What do these formulas for S(T) and N(T) predict for the length of searching period and percentage of larvae surviving the searching period when the air temperature is 25 degrees Celsius?

Part 2) Sketch the graph of N(T), and determine the temperature at which the largest percentage of codling moth larvae survive. Then determine the temperature at which the smallest percentage of larvae survive. (Remember, 20 ≤ T ≤ 30.)

Part 3) Find the rate of change of the searching period with respect to temperature T. When does this rate equal zero? What (if anything) occurs when?

Part 4) Find the rate of change of the percentage of larvae surviving the searching period with respect to the length of the searching period using the chain rule, What (if any) information does this rate provide.