Problem 44 - Precalculus

Question

A sliding door is closed by pulling a cord with a constant force of 45 pounds at a constant angle of \(55^{\circ}\) The door is moved 6 feet to close it. How much work is done?

Step-by-Step Solution

Answer

The work done is approximately 154.2 foot-pounds.

Step 1: Understand Work Formula

Work is calculated using the formula \( W = F \, d \, \cos(\theta) \), where \( W \) is the work done, \( F \) is the force applied, \( d \) is the distance over which the force is applied, and \( \theta \) is the angle between the force and the direction of motion. In this problem, the force \( F = 45 \) pounds, the angle \( \theta = 55^{\circ} \), and the distance \( d = 6 \) feet.

Step 2: Calculate Cosine of the Angle

Find \( \cos(55^{\circ}) \) using a calculator to determine how much of the force is used in the direction of movement. \( \cos(55^{\circ}) \approx 0.5736 \).

Step 3: Plug Values into Equation

Substitute the given values and calculated cosine into the work formula: \[ W = 45 \, \text{pounds} \times 6 \, \text{feet} \times \cos(55^{\circ}) \].

Step 4: Perform the Calculation

Calculate the expression: \( W = 45 \times 6 \times 0.5736 = 154.152 \).

Step 5: Provide the Final Answer

Round the result if necessary. Therefore, the work done to close the door is approximately 154.2 foot-pounds.

Key Concepts

Force and Motion

In physics, force and motion are fundamental concepts that describe how objects interact in our world. Force is any interaction that, when unopposed, will change the motion of an object. For example, if you push a door, you're applying a force to make it move. Motion, on the other hand, refers to the movement of an object from one place to another.

• Force: Measured in units like pounds or newtons, it causes an object to accelerate or change its state of motion.
• Motion: Described in terms of displacement, velocity, and acceleration, which are dependent on forces acting on the object.

When we talk about work in physics, we're closely examining how force causes motion. The work done by a force depends on the magnitude of the force applied, the distance over which it is applied, and the direction of the force relative to the direction of motion. In the case of the sliding door, a force of 45 pounds is applied to close it over a distance of 6 feet, and the angle plays a crucial role in determining how effective this force is.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles of triangles. It is particularly useful in physics for resolving components of forces when they aren't applied in a straight line.

• Triangles: The main focus is right triangles, in which the relationship of angles often involves sine, cosine, and tangent functions.
• Angle Measurements: Typically measured in degrees or radians, crucial for calculating sides of triangles.

In our scenario with the sliding door, the angle of the applied force is crucial. It is measured as 55 degrees from the horizontal direction of motion. By using trigonometry, specifically the cosine function, we can determine the component of the force that effectively contributes to the door's movement.

Cosine Function

The cosine function is a fundamental trigonometric function. It relates the angle of a triangle to the ratio of the adjacent side to the hypotenuse. In physics, it's often used to determine how much of a force acts in a particular direction.

• Cosine of an Angle: Denoted as \( \cos(\theta) \), where \( \theta \) is the angle. For an angle of 55 degrees, \( \cos(55^{\circ}) \approx 0.5736 \).
• Application: When solving for work, \( \cos(\theta) \) helps isolate the effective force directly applied in the direction of motion.

In the problem of the sliding door, calculating \( \cos(55^{\circ}) \) as 0.5736 allows us to determine how much of the 45-pound force is efficiently closing the door. By multiplying this cosine value with the force and distance, we find the total work done to be 154.2 foot-pounds, providing a clear measure of the energy used in this physical task.