Illustrate a detailed image of a kitchen scenario. Visualize a large, steaming tray of lasagna being taken out from an oven. Its melted cheese and rich tomato sauce give off a mouth-watering aroma. The oven should be shown at 200°F and the room temperature should be clearly indicated as 70°F. The kitchen should be well-equipped, with pots and pans hanging, a chopping board with vegetables on, and a table, on which the tray of lasagna is being placed. A clock on the wall should be shown to emphasize the element of time with the hands pointing at different hours.

A tray of lasagna comes out of the oven at 200°F and is placed on a table where the surrounding room temperature is 70°F. The temperature T (in °F) of the lasagna is given by the function T(t) = e^(4.86753 - t) + 70, where t is time (in hours)after taking the lasagna out of the oven. t is greater than or equal to 0.

At which of the following times is the lasagna cooling SLOWEST?

A. 3 hours after being taken out of the oven
B. 5 hours after being taken out of the oven
C. 6 hours after being taken out of the oven
D. 7 hours after being taken out of the oven
E. 9 hours after being taken out of the oven

I want to say that it is E. 9 hours after being taken out of the oven, but I am a little unsure. Any help is appreciated!

subbing in the time will give you the temperature.

e.g. I will do B
T(5) = e^(4.86753-5) + 70
= 70.88°

and for E:
T(9) = e^(4.85753-9) + 70 = 70.016°

Your last question deals with the "rate of change"
You can use common sense or intuition to tell that the slowest change would happen at 9 hrs

To find exact answers, take the derivative of your equation, and sub in the different values.

Its 9

Well, let's take a closer look at the function T(t) = e^(4.86753 - t) + 70. This equation represents the temperature T of the lasagna at time t after being taken out of the oven.

To determine when the lasagna is cooling slowest, we need to find when the rate of change of temperature is the smallest. This is equivalent to finding when the derivative of the temperature with respect to time is the smallest.

Let's find the derivative of T(t) and solve for when it equals zero. We have:

T'(t) = -e^(4.86753 - t)

Setting T'(t) = 0, we get:

-e^(4.86753 - t) = 0

Since the exponential function is never zero for any real value of t, we can conclude that there is no time at which the lasagna is cooling slowest.

So, the answer is actually none of the above options. The lasagna is not cooling slowest after any specific time.

But hey, I can tell you a lasagna joke instead! Why did the lasagna go to the spa? It wanted to relax its noodle!

To find the time at which the lasagna is cooling slowest, we need to determine the maximum value of the derivative of the temperature function T(t). The derivative of T(t) can be found by differentiating the function with respect to t:

T'(t) = -e^(4.86753 - t)

To find the maximum value of T'(t), we set T'(t) equal to zero and solve for t:

-e^(4.86753 - t) = 0

Since exponential functions are never equal to zero, there is no solution to this equation. Therefore, there is no time at which the lasagna is cooling slowest. The correct answer is none of the given choices.

To determine at which time the lasagna is cooling the slowest, we need to find the point where the derivative of the temperature function T(t) is equal to zero.

First, let's find the derivative of T(t):

T'(t) = (e^(4.86753 - t))' = -e^(4.86753 - t)

To find the time when the lasagna is cooling the slowest, we need to find the t-value that makes T'(t) = 0. So let's solve for t:

- e^(4.86753 - t) = 0

Since the exponential function e^x is never equal to zero for any real value of x, this equation has no solution. Therefore, the lasagna never cools down completely, and there is no time when it is cooling the slowest.

To clarify, the correct answer is none of the given options (A, B, C, D, E).