Problem 38 - College Algebra

Question

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

Hurricane shutters should be installed as the window is expected to experience 540 pounds of force, which is more than what it can withstand (300 pounds)

Step 1: Establish the formula for joint variation

For joint variation, the formula is generally \(y = kxz\). In this case, the force of wind (F) varies jointly as the area of the window (A) and the square of the wind's speed (v^2). So the formula will be \(F = k \cdot A \cdot v^2\). Here, \(k\) is a constant that we need to find first using provided numbers.

Step 2: Calculate the constant of variation

The values given are a wind of 30 miles per hour on a window measuring 4 feet by 5 feet equals a force of 150 pounds. So, we get \(150 = k \cdot (4 \times 5) \cdot (30^2)\). Solving this, we get \( k = 0.025 \).

Step 3: Find the force in the storm with given speed and window area

Now, we can use the value for \(k\) in the formula to calculate the force due to the storm winds. The storm winds are blowing at 60 miles per hour and the window measures 3 feet by 4 feet. So, we get \( F = 0.025 \cdot (3 \cdot 4) \cdot (60^2)\). Solving this, we get \(F = 540\) pounds.

Step 4: Decision to place hurricane shutters

The calculated force (540 pounds) is more than the window can withstand (300 pounds). So, yes, hurricane shutters should definitely be placed on the window.