Problem 1 - Technical Mathematics with Calculus

Question

The temperature \(T\) inside a certain furnace is described by the equation \(T=55.6 t^{2}+28.2 t+44.8^{\circ} \mathrm{F},\) where \(t\) is the elapsed time in hours. Find the time rate of change of the temperature at \(t=2.00 \mathrm{h}\)

Step-by-Step Solution

Answer

The time rate of change of the temperature at t=2.00 h is 250.6 degrees Fahrenheit per hour.

Step 1: Identify the Function to Differentiate

Recognize that the temperature inside the furnace, represented by the function T with respect to time t, is a polynomial. The function is given by T(t) = 55.6t^2 + 28.2t + 44.8.

Step 2: Differentiate with respect to Time

Compute the derivative of T with respect to time t to find the time rate of change of the temperature. The derivative T'(t) is calculated by differentiating each term separately: T'(t) = d/dt(55.6t^2) + d/dt(28.2t) + d/dt(44.8).

Step 3: Calculate the Derivative

By applying the power rule to each term, we get T'(t) = 2*55.6t + 28.2, because the derivative of a constant is 0 and does not appear in the final expression.

Step 4: Evaluate the Derivative at t = 2.00 hours

Substitute t = 2.00 into the derivative to find the time rate of change of the temperature at that specific time: T'(2.00) = 2*55.6*2.00 + 28.2.

Step 5: Compute the Result

Calculate the numeric value of the derivative at t = 2.00, which gives T'(2.00) = 2*55.6*2.00 + 28.2 = 222.4 + 28.2 = 250.6 degrees Fahrenheit per hour.

Key Concepts

Derivative Calculation

Understanding derivative calculations is crucial in calculus as it allows us to find the rate at which a function is changing at any given point. For example, take the case of the temperature function inside a furnace over time, defined as T with respect to time t. To find the rate at which the temperature changes, we first need to perform a derivative calculation of the temperature function with respect to time.

The process involves applying calculus rules to obtain the function's derivative, denoted as T'(t) or dT/dt. This derivative represents the instantaneous rate of change of the function, much like how the speedometer in a car shows the instantaneous speed. After obtaining the derivative as a function of time, we can evaluate it at a specific time to understand the temperature change rate at that precise moment.

Polynomial Function Differentiation

Differentiating polynomial functions is a fundamental concept in calculus, especially as it applies to physical scenarios like changes in temperature over time. The provided function T(t) = 55.6t^2 + 28.2t + 44.8 is a polynomial, and its differentiation is done term-by-term by applying the power rule.

Power Rule for Differentiation

For any term in the form at^n, where a is a constant and n is the power, the derivative is found by multiplying the coefficient a by the power n, and reducing the power by one. The differentiation of a constant term, like 44.8 in our temperature function, is zero since constants do not change.

When we apply these rules to the given polynomial, we see that the rate at which the temperature changes is influenced by both the square of the time (t^2) and the time itself (t). The constant (44.8) represents the initial temperature, which does not change over time and thus has no impact on the rate.

Time Rate of Change

The time rate of change in a physical context, such as the rate of temperature change in a furnace, is essential for understanding how quickly conditions are evolving. After differentiating the polynomial that describes temperature over time, the evaluation of this derivative at a specific point gives the time rate of change at that particular moment.

In our scenario, finding the time rate of change at t = 2.00 hours involves substituting this value into the derivative function. This computation tells us how fast the temperature is rising or falling within the furnace at the two-hour mark. This information can be critical in processes that depend on precise temperature controls, such as manufacturing or baking. Calculating the time rate of change provides a snapshot of the dynamic process happening inside the furnace, helping us predict and regulate temperature effectively.