Read each problem carefully and you must show all work for full credit.

Preparing A Picnic
Is that picnic food safe to eat? That depends. At certain temperature ranges, some foods are breeding grounds for harmful bacteria. How can an exponential function help you prepare picnic food that is bacteria-free?

Newton’s Law of Cooling models the temperature of a cooling body. The model is the sum of a decaying exponential function and a constant function. It defines the current temperature T of a body as where TR is the surrounding temperature, T0 is the initial temperature of the body, r is a cooling constant, and t is the time (in minutes) the body has been sitting. This law can help you make decisions about the safety of your food.

You are preparing a picnic of roasted turkey sandwiches, ham salad, broccoli quiche, and tomato soup in a kitchen that is 70º. For these foods, the cooling constant r is 0.046. You have cooked each dish according to food safety standards. The following table gives the temperature as the food comes out of the oven and off of the stove.

Food
Temperature (Fº)
Roast Turkey
165°
Baked Ham
140°
Bacon Quiche
160°
Tomato Soup
212°

Complete the table using the formula given above to show the time before each food reaches 140º where bacteria will begin to grow.

Food
Time (min) before growth begins
Roast Turkey

Bacon Quiche

Tomato Soup

Turkey
T= (165-70)e^(-0.046*t)+70
t= (95)e^(-0.046*t)+70

If we allow the food to get to room temperature, 73º, then some bacteria may begin to double. Use the same formula changing the current temperature, T, to 73º to calculate the time before bacteria begins to double in each food listed below in the table.

Food
Time (min) before doubling begins
Roast Turkey

Baked Ham

Denis is looking at the weather for the picnic and is looking over some data regarding the temperatures and how the temperature relates to the number of mosquitoes. The data are represented by the exponential function x = log381 where x is the number of mosquito populations within a square foot when the temperature is 81º. Convert this equation to an exponential function and solve for x. What does this tell us about the number of mosquitoes that could be out during our picnic?

Please show your work (I was gonna set high pts but ik someone will just steal them)