Problem 75 - Precalculus: Functions and Graphs

Question

The temperature \(T\) within a cloud at height \(h\) (in feet) above the cloud base can be approximated using the equation \(T=B-\left(\frac{3}{1000}\right) h,\) where \(B\) is the temperature of the cloud at its base. Determine the temperature at \(10,000\) feet in a cloud with a base temperature of \(55^{\circ} \mathrm{F}\) and a base height of 4000 feet. Note: For an interesting application involving the three preceding exercises, see Exercise 14 in the Discussion Exercises at the end of the chapter.

Step-by-Step Solution

Answer

The temperature at 10,000 feet is 25°F.

Step 1: Plug in Known Values

To find the temperature at a certain height within the cloud, you need to input the known values into the given formula. You are given:- The temperature at the cloud base, \(B = 55^{\circ} \mathrm{F}\)- The height \(h\) in the cloud is \(10,000\) feet.The formula for temperature at a height is: \(T = B - \left(\frac{3}{1000}\right)h\).

Step 2: Calculate the Temperature Reduction

First, calculate the reduction in temperature due to the height.The reduction is given by \(\left(\frac{3}{1000}\right)h\).Substitute \(h = 10,000\) into the equation:\[\left(\frac{3}{1000}\right) \times 10,000 = 30\] degrees.

Step 3: Subtract to Find the Temperature

Subtract the calculated temperature reduction from the base temperature to get the cloud temperature at 10,000 feet.Base Temperature, \(B = 55^{\circ} \mathrm{F}\)Reduction = 30 degrees\[T = 55 - 30 = 25^{\circ} \mathrm{F}\]

Key Concepts

Temperature Gradient

The temperature gradient refers to how temperature changes as you move to different heights within the cloud. In this scenario, the temperature gradient is described by the equation \( T = B - \left(\frac{3}{1000}\right)h \). This formula indicates that as you increase altitude, the temperature decreases at a specific rate.

To put it simply, the factor \( \left(\frac{3}{1000}\right) \) represents the rate at which temperature drops per 1000 feet increase in height. Therefore, every time you ascend 1000 feet in the cloud, you can expect the temperature to drop by 3 degrees Fahrenheit.

• The constant \( \frac{3}{1000} \) signifies 3 degrees Fahrenheit reduction per 1000 feet.
• This value determines how quickly the temperature changes with elevation in the cloud.

Height and Temperature Relationship

The relationship between height and temperature within clouds can be directly modeled through a linear equation. As mentioned earlier, the mathematical equation \( T = B - \left(\frac{3}{1000}\right)h \) allows us to forecast the temperature at different altitudes by observing the change from the base.

The term \( B \) in this equation is crucial. It acts as the starting temperature, or the temperature at the base of the cloud. From there, as height \( h \) increases, temperature \( T \) decreases due to the negative gradient factor.

• Higher altitudes result in lower temperatures.
• The initial base temperature is reduced by an amount proportional to the height.

For our specific example, height increases from 4000 to 10000 feet, leading to a 30-degree Fahrenheit drop in temperature. Understanding this fundamental relationship is vital to calculating temperatures at various altitudes within a cloud.

Temperature at Cloud Base

The temperature at the cloud base is a key reference point when determining temperatures at higher points within the cloud. In the provided example, this base temperature \( B \) is set at 55 degrees Fahrenheit.

This base temperature is not just an initial condition for our calculations, but also a critical parameter determining the temperature profile of the cloud itself. The importance of knowing the cloud base temperature becomes evident when solving for higher altitudes; without it, the temperature estimation would be impossible.

• \( B = 55^{\circ} \mathrm{F} \) in our example, setting the starting point for the calculations.
• It impacts all subsequent temperature calculations as we move upwards in the cloud layer.

Using the base temperature, we successfully derived a temperature of 25 degrees Fahrenheit at 10,000 feet by factoring in both this initial condition and the temperature gradient.