Problem 50 - Calculus Single Variable

Question

The heat index is the apparent temperature \(T\left({ }^{\circ} \mathrm{F}\right)\) that a human feels. A rough aproximation of \(T\) in terms of actual temperature \(x\left({ }^{\circ} \mathrm{F}\right)\) and relative humidity \(h,\) where \(h\) is a number in the interval \([0,100],\) is \(T=2.7+0.885 x-\) \(0.787 h+0.012 x h .\) Suppose that the actual temperature is increasing at the rate of \(1.7^{\circ} \mathrm{F} / \mathrm{hr}\) at the moment \(x=88\) and \(h=60 .\) How fast must humidity decrease for the heat index to be unchanged?

Step-by-Step Solution

Answer

Humidity must decrease at about -2.518 percent per hour.

Step 1: Understand the given formula

We are given the formula for the heat index: \( T = 2.7 + 0.885x - 0.787h + 0.012xh \). The problem also states the actual temperature rate of change as \( \frac{dx}{dt} = 1.7 \frac{\degree F}{\text{hr}} \). We need to find \( \frac{dh}{dt} \) when \( x=88 \) and \( h=60 \) such that \( \frac{dT}{dt} = 0 \).

Step 2: Set dT/dt to zero

To keep the heat index unchanged, we need the rate of change of \( T \) with respect to time to be zero: \( \frac{dT}{dt} = 0 \).

Step 3: Differentiate T with respect to time

Differentiate the heat index formula with respect to time using the chain rule: \( \frac{dT}{dt} = 0.885 \frac{dx}{dt} + (0.012x - 0.787) \frac{dh}{dt} + 0.012h \frac{dx}{dt} \).

Step 4: Substitute given values

Substitute the known values into the differentiated equation: \( 0 = 0.885(1.7) + (0.012(88) - 0.787) \frac{dh}{dt} + 0.012(60)(1.7) \).

Step 5: Solve for dh/dt

Simplify and solve for \( \frac{dh}{dt} \):1- Calculate \( 0.885 \times 1.7 \).2- Calculate \( 0.012 \times 88 - 0.787 \).3- Calculate \( 0.012 \times 60 \times 1.7 \).4- Solve the equation for \( \frac{dh}{dt} \).This will result in \( \frac{dh}{dt} \approx -2.518 \).

Step 6: Conclusion

For the heat index to remain unchanged, the humidity must decrease at a rate of approximately \(-2.518\) percentage points per hour.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It involves finding the rate at which a quantity changes over time or in relation to another quantity. In simple terms, differentiation is about calculating derivatives. These derivatives tell you how a function changes as its input changes.

In the context of this problem, differentiation was used to determine how the heat index (\( T \)) changes with respect to time. The heat index depends on actual temperature (\( x \)) and humidity (\( h \)). By differentiating the heat index formula with respect to time, we can analyze how changes in temperature and humidity affect the heat index.
The formula derived from differentiation here is:

• \( \frac{dT}{dt} = 0.885 \frac{dx}{dt} + (0.012x - 0.787) \frac{dh}{dt} + 0.012h \frac{dx}{dt} \)

This helps find the necessary rate of change for the variable of interest, in this case, humidity.

Chain Rule

The chain rule is a crucial method in calculus for differentiating composite functions. It links several rates of change together. Simply put, if one variable depends on another through a chain of relationships, the chain rule helps find the derivative across this chain.

In this exercise, you encountered a function for the heat index, which depends on both temperature and humidity. However, these vary over time. The chain rule allowed us to consider the changes in both temperature and humidity concerning time.
Applying the chain rule here segmented the problem into manageable pieces:

• Differentiate each variable separately.
• Multiply the derivative by the rate at which each variable changes with respect to time.

This led to the composite derivative used to calculate the necessary decrease in humidity to keep the heat index unchanged.

Rate of Change

The rate of change is a term used to describe how one quantity changes relative to another. It's a key concept in understanding any dynamic system. In this problem, the focus is on how the heat index remains constant as the temperature and humidity change.

Given the exercise, you had to ensure the heat index did not change. To achieve this, you needed to calculate the rate at which humidity had to decrease to balance out the increase in temperature. In the step-by-step solution, the given rate of temperature change was \( \frac{dx}{dt} = 1.7 \) degrees per hour. You needed to find \( \frac{dh}{dt} \), the rate of humidity change.

By solving the differentiated equation:

• You equated the total rate of change of the heat index to zero.
• This allowed for isolating \( \frac{dh}{dt} \) and solving mathematically.

The result showed that humidity must decrease by approximately \(-2.518\) percentage points per hour for the heat index to remain constant.