Problem 2 - Algebra and Trigonometry

Question

Use the model given to answer the questions about the object or process being modeled. A mountain climber models the temperature \(T\left(\text { in }^{\circ} \mathrm{F}\right)\) at levation \(h(\text { in } \mathrm{ft})\) by $$ T=70-0.003 h $$ (a) Find the temperature \(T\) at an elevation of 1500 \(\mathrm{ft}\) . (b) If the temperature is \(64^{\circ} \mathrm{F}\) , what is the elevation?

Step-by-Step Solution

Answer

(a) The temperature at 1500 ft is 65.5°F. (b) The elevation is 2000 ft when the temperature is 64°F.

Step 1: Identify Given Information for Part (a)

For part (a), we need to find the temperature \(T\) at a given elevation of 1500 ft. Given is the equation of temperature as a function of elevation: \(T = 70 - 0.003h\), where \( h = 1500 \text{ ft} \).

Step 2: Plug the Elevation into the Equation

Substitute \( h = 1500 \) into the equation: \[ T = 70 - 0.003 \times 1500 \] Calculate the product to find the value of \(T\).

Step 3: Calculate the Temperature at 1500 ft

First multiply the constant change rate by the elevation: \(0.003 \times 1500 = 4.5\). Now, calculate the temperature: \[ T = 70 - 4.5 = 65.5^{\circ} \text{F} \].

Step 4: Identify Given Information for Part (b)

For part (b), we need to find the elevation \( h \) when the temperature is given as \(64^{\circ} \text{F}\). The equation is: \( T = 70 - 0.003h \). Set \( T = 64 \).

Step 5: Rearrange the Equation to Solve for Elevation

Rearrange the equation to solve for \( h \): \[ 64 = 70 - 0.003h \]. By subtracting 64 from 70, we find \( 70 - 64 = 6 \). Now, we solve \( 6 = 0.003h \).

Step 6: Solve for Elevation

To isolate \( h \), divide both sides by \(0.003\): \( h = \frac{6}{0.003} \). Calculate \( h \) to find the elevation.

Step 7: Calculate the Elevation

Divide to find \( h \): \[ h = \frac{6}{0.003} = 2000 \]. This means the elevation is 2000 ft when the temperature is \(64^{\circ} \text{F}\).

Key Concepts

Linear Equation

Linear equations are mathematical expressions that create a straight line when graphed. They consist of constants, coefficients, and variables. In the exercise, the linear equation used is:
\[T = 70 - 0.003h\]This equation allows us to calculate the temperature at a given elevation. Here, the variable \(h\) represents the elevation in feet, while \(T\) expresses the temperature in degrees Fahrenheit. The number \(70\) is a constant, which likely represents the temperature at sea level, and \(-0.003\) is the coefficient indicating the temperature's rate of change as altitude increases.
Understanding this relationship is crucial, especially for situations like mountain climbing, where temperatures decrease as you ascend.

Temperature Calculation

Temperature calculation using a linear equation involves substituting the elevation into the equation to find the result. For instance, when the elevation is 1500 ft, you plug this value into the equation to find \(T\):
\[T = 70 - 0.003 imes 1500\]Carrying out the multiplication gives \(4.5\), which we subtract from \(70\), resulting in a temperature of \(65.5^{\circ} \mathrm{F}\).

• First, identify the given elevation or temperature.
• Substitute the known value into the formula.
• Solve the equation to find the unknown variable.

This process shows how mathematical modeling can be used to predict real-world phenomena, such as how temperature changes with elevation.

Elevation Calculation

Elevation calculation involves working backward from known temperatures. If given the temperature, such as \(64^{\circ} \mathrm{F}\), we can find the elevation using the equation:
\[64 = 70 - 0.003h\]Rearrange the equation to isolate \(h\):
\[6 = 0.003h\]By dividing both sides by \(0.003\), we determine:\[h = \frac{6}{0.003} = 2000\]This calculation tells us that at an elevation of 2000 ft, the temperature is \(64^{\circ} \mathrm{F}\). This demonstrates how we can determine one variable when the other is known, highlighting the practical application of linear equations in real-world scenarios.

Mountain Climbing

In mountain climbing, understanding temperature and elevation is critical. As climbers ascend, the temperature typically drops, as seen with the given linear model. This is significant because lower temperatures at higher elevations can affect not only comfort but also safety.

• Temperature impacts physical exertion and health.
• Hypothermia becomes a risk in cold climates.
• Knowledge of elevation helps in navigation and planning.

Using linear equations as part of mathematical modeling enables climbers to better anticipate and prepare for changing conditions. This preparation can lead to more successful, safer climbs.