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, 0 is less than or equal to t, where t is time (in hours)after taking the lasagna out of the oven. What is the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven?
-17.59
dT/dt = -130 e^-t
so plug in t = 2
Well, I'm not really an expert on lasagna physics, but let's give it a shot! According to the given function, T(t) = e^(4.86753 - t) + 70, we need to find the rate of change of temperature, which is the derivative of T(t) with respect to time (t). So let's differentiate T(t):
dT/dt = -e^(4.86753 - t)
Now, let's plug in t = 2 to find the rate of change exactly 2 hours after taking the lasagna out:
dT/dt = -e^(4.86753 - 2)
= -e^(2.86753)
So, the rate of change in temperature exactly 2 hours after taking it out of the oven is approximately -e^(2.86753) degrees Fahrenheit per hour. But remember, I'm just a clown bot, so take this result with a pinch of salt!
To find the rate of change in temperature of the lasagna exactly 2 hours after taking it out of the oven, we need to find the derivative of the temperature function T(t) with respect to time.
The temperature function is given as T(t) = e^(4.86753 - t) + 70.
Taking the derivative of T(t) with respect to t using the chain rule, we get:
dT/dt = -e^(4.86753 - t).
Now, we can evaluate the derivative at t = 2 to find the rate of change at that time:
dT/dt at t = 2 = -e^(4.86753 - 2).
Calculating this value, we get:
dT/dt at t = 2 = -e^(2.86753).
So, the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven is approximately -e^(2.86753) °F/hour.
To find the rate of change in the temperature of the lasagna exactly 2 hours after taking it out of the oven, we need to differentiate the function T(t) with respect to time t.
The given function for the temperature of the lasagna is T(t) = e^(4.86753 - t) + 70.
Differentiating T(t) with respect to t will give us the rate of change of the temperature.
dT/dt = d/dt (e^(4.86753 - t) + 70)
To differentiate this function, we use the chain rule of differentiation. According to the chain rule, if we have a function f(g(t)), the derivative of f(g(t)) with respect to t is f'(g(t)) * g'(t).
Let's apply the chain rule to our function:
dT/dt = d/dt (e^(4.86753 - t)) + d/dt (70)
The derivative of e^(4.86753 - t) with respect to t is (-1) * e^(4.86753 - t) because the derivative of e^x with respect to x is e^x.
Therefore, the rate of change in the temperature of the lasagna can be calculated as:
dT/dt = (-1) * e^(4.86753 - t)
To find the rate of change exactly 2 hours after taking the lasagna out of the oven, substitute t = 2 into the equation:
dT/dt = (-1) * e^(4.86753 - 2)
Now, we can calculate the rate of change dT/dt by evaluating the expression.