The heat of respiration of leafy greens as a function of...

The heat of respiration of leafy greens as a function of time when the temperature is changing during a cooling operation is as follows: $$ q=\int_0^t 0.009854[e]^{0.073 T} d t $$ where $\mathrm{q}$ is in $\mathrm{BTU} /(\mathrm{b}), \mathrm{T}$ is in ${ }^{\circ} \mathrm{F}$, and $\mathrm{t}$ is time in hours. The temperature of a box of spinach containing $50 \mathrm{lb}$, originally at $110^{\circ} \mathrm{F}$, will cool down exponentially when placed in a refrigerated room at $35^{\circ} \mathrm{F}$ according to: $$ \mathrm{T}=35+(75) \exp (-\mathrm{t} / 5) $$ where $\mathrm{t}$ is the time in hours. Evaluate this integral using a spreadsheet and determine the total heat generated by the spinach as the box cools down from $110^{\circ} \mathrm{F}$ to $40^{\circ} \mathrm{F}$. Use Simpson's rule to determine the value of the integral.

The heat of respiration of leafy greens as a function of time when the temperature is changing during a cooling operation is as follows:
$$
q=\int_0^t 0.009854[e]^{0.073 T} d t
$$
where $\mathrm{q}$ is in $\mathrm{BTU} /(\mathrm{b}), \mathrm{T}$ is in ${ }^{\circ} \mathrm{F}$, and $\mathrm{t}$ is time in hours. The temperature of a box of spinach containing $50 \mathrm{lb}$, originally at $110^{\circ} \mathrm{F}$, will cool down exponentially when placed in a refrigerated room at $35^{\circ} \mathrm{F}$ according to:
$$
\mathrm{T}=35+(75) \exp (-\mathrm{t} / 5)
$$
where $\mathrm{t}$ is the time in hours. Evaluate this integral using a spreadsheet and determine the total heat generated by the spinach as the box cools down from $110^{\circ} \mathrm{F}$ to $40^{\circ} \mathrm{F}$. Use Simpson's rule to determine the value of the integral.

Key Concepts

Spreadsheet Applications in Numerical Computation

Spreadsheets can be employed as effective tools for numerical computations, including the evaluation of integrals using methods like Simpson’s rule. They provide a user-friendly, accessible platform for setting up iterative calculations, organizing data, and visualizing results, which makes them particularly useful for practical problem-solving in both academic and professional settings.

Exponential Functions

Exponential functions describe processes that change at rates proportional to their current value, and they are common in both growth and decay phenomena. Their properties are critical in modeling real-world scenarios such as chemical reactions, population dynamics, and thermal processes, and understanding how to integrate these functions is a key aspect of calculus.

Thermal Cooling Models

Thermal cooling models, such as those based on exponential decay, represent how the temperature of an object changes over time, often following Newton's law of cooling. These models are essential in engineering and science for predicting temperature changes in various materials and systems, and they are frequently coupled with other physical responses, like heat generation or loss.

Numerical Integration

Numerical integration is a computational method used to approximate the value of definite integrals when an analytic solution is difficult or impossible to obtain. It involves breaking the integral's domain into small segments and summing the contributions of these segments, offering a practical approach for evaluating integrals of functions that may be complex or only known discretely.

Simpson's Rule

Simpson's rule is a specific technique in numerical integration that approximates the integrand using quadratic (parabolic) segments over subintervals. It generally yields highly accurate estimates with relatively few intervals, making it a popular method for integrating smooth functions, especially when high precision is required in engineering and physical sciences.

Transcript

00:10 The temperature of food placed in a refrigerator after t hours is approximated by this function.

00:20 So to find out what the temperature is, after being in the refrigerator a certain length of time, or t hours, we just put the hours into the function everywhere you see t and then evaluate.

00:38 So for instance, for t equals zero, we simply put in zero everywhere we see t and evaluate.

00:49 So here we are multiplying this by zero, this is zero, this is zero, this is zero.

01:00 So this one is pretty easy.

01:02 We end up with 75 over 10 times 10.

01:08 Don't forget this 10 in front, which multiplies out.

01:15 So we end up with a temperature of 75 degrees.

01:20 And that is our starting temperature when time equals zero.

01:24 That's the temperature of the food when it was first put into the refrigerator.

01:29 Now to see the temperature of the food after two hours, again, just put in to everywhere you see t and evaluate.

01:41 Be careful that you follow the order of operations when you're evaluating these.

01:45 So we need to do the exponents and then the multiplication and then the addition.

01:56 So we treat the numerator separately from the denominator.

02:01 So we have four times two squared, which is four times four, plus 16 times two, which is 32, plus 75.

02:15 And then in the bottom we have two squared, which is four, plus 75.

02:20 Plus 4 times 2, which is 8, plus 10.

02:26 So we get 10 times all of this.

02:29 16 plus 32 plus 75.

02:33 So 16 plus 32 is 48 plus 75, so 123.

02:44 So 123.

02:45 And then 4 plus 8 plus 10, so 12 plus 10 is 22...

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