Problem 38 - Algebra and Trigonometry

Question

Use the four-step procedure for solving variation problems given on page 445 to solve Exercises 21–36. The force of wind blowing on a window positioned at a right angle to the direction of the wind varies jointly as the area of the window and the square of the wind’s speed. It is known that a wind of 30 miles per hour blowing on a window measuring 4 feet by 5 feet exerts a force of 150 pounds. During a storm with winds of 60 miles per hour, should hurricane shutters be placed on a window that measures 3 feet by 4 feet and is capable of withstanding 300 pounds of force?

Step-by-Step Solution

Answer

Yes, hurricane shutters should be placed on the window that measures 3 feet by 4 feet. This is because the force exerted on it during the storm is estimated at approximately 359.88 pounds, which exceeds the window's maximum withstandable force of 300 pounds.

Step 1: Determine the Joint Variation Equation

First things first, using the words of the problem 'varies jointly as', this can be translated into a joint variation equation: \( F = kAW^2 \), where: - \( F \) is the force on the window - \( A \) is the area of the window - \( W \) is the wind speed - \( k \) is the constant of variation

Step 2: Compute the Constant of Variation

Given that a wind of 30 mph on a window of 4 feet by 5 feet exerts a force of 150 pounds, we can substitute these values into the equation. The area of the window is the product of the length and width, so A = 4 feet * 5 feet = 20 square feet. Thus, we have \( 150 = k * 20 * 30^2 \). Solving for \( k \), we get \( k = 150 / (20 * 30^2) = 0.00833 \) pounds/(square foot*(miles/hour)^2).

Step 3: Compute the Force Withstood by the New Window

Now we know that the storm has wind speeds of 60 mph. We need to calculate the force exerted on the new window, which has an area of 3 feet by 4 feet, or 12 square feet. Substituting these values into the equation gives us \( F = 0.00833 * 12 * 60^2 = 359.88 \) pounds.

Step 4: Determine if the New Window Needs Shutters

Now, we need to compare the force exerted on the window during the storm (359.88 pounds) with the maximum force the window can withstand (300 pounds). Since 359.88 pounds > 300 pounds, the window would not withstand the storm and hurricane shutters need to be placed as a protective measure.

Key Concepts

Joint Variation Equation

In algebra, the joint variation equation is used to describe situations where a quantity varies in direct proportion to the product of two or more other quantities. Here's an example: If the force exerted by the wind on a window varies jointly as the area of the window and the square of the wind speed, we can describe this relationship with the equation \( F = kAW^2 \).

This equation tells us that the force \( F \) is proportionally related to the area \( A \) of the window and the square of the wind speed \( W \). These types of equations are crucial for understanding how different factors can influence a particular outcome, like the force on a window during a windy day. By incorporating the constant of variation \( k \), we adjust the equation to fit real-world observations.

Constant of Variation

The constant of variation, represented by \( k \) in our equation, is what ties together the quantities that are varying jointly. It is the value that computes the proportionality of the relation. For instance, when we find that a 30 mph wind exerts a force of 150 pounds on a window with an area of 20 square feet, we derive the constant \( k \) by rearranging the equation and solving for \( k \), resulting in \( k = 0.00833 \).

This constant is a critical part of the equation because it scales the relationship to match real-world data. Once we've figured out this constant, we can use it to predict other outcomes, such as the force on different sized windows or under different wind conditions.

Wind Force Calculation

Calculating wind force on structures like windows is essential for safety and engineering purposes. Using the relationship set by the joint variation equation, we can predict the force on any window if we know the area and wind speed. For our textbook problem, we found the force on a new window to be \( F = 0.00833 \times 12 \times 60^2 = 359.88 \) pounds.

Educational exercises like this one not only help us understand abstract mathematical concepts but also apply them to real-world scenarios like estimating the impact of high winds during storms. This kind of practical application is why mastering variation equations is so valuable.

Mathematical Problem-Solving

Mathematical problem-solving involves several key steps, starting with understanding the problem, identifying the right techniques to use, performing the mathematical calculations, and then interpreting the results. In our scenario, by understanding joint variation and the constant of variation, we're able to approach the problem methodically, calculate wind force, and make a safety decision.

The four-step procedure applied here is typical of mathematical problem-solving and underlines the importance of a structured approach in finding solutions to real-world problems. It showcases the power and necessity of mathematics in making informed, logical decisions based on quantitative data.