Problem 30 - Algebra and Trigonometry

Question

Computing Work A wagon is pulled horizontally by exerting a force of 20 pounds on the handle at an angle of \(30^{\circ}\) with the horizontal. How much work is done in moving the wagon 100 feet?

Step-by-Step Solution

Answer

The work done is \(1000 \sqrt{3}\) foot-pounds.

Step 1: Identify the Formula for Work

The formula to calculate work is given by: \[ W = F \times d \times \text{cos}(\theta) \] where: - \( W \) is the work done, - \( F \) is the force applied, - \( d \) is the distance moved, - \( \theta \) is the angle between the force and the direction of movement.

Step 2: Substitute the Given Values

Identify the values given in the problem: - Force \( F = 20 \) pounds - Distance \( d = 100 \) feet - Angle \( \theta = 30^{\theta} \) Substitute these values into the formula: \[ W = 20 \times 100 \times \text{cos}(30^{\theta}) \]

Step 3: Compute the Cosine of the Angle

Calculate the cosine of the angle \( 30^{\theta} \): \[ \text{cos}(30^{\theta}) = \frac{\sqrt{3}}{2} \] Substitute this value into the equation: \[ W = 20 \times 100 \times \frac{\sqrt{3}}{2} \]

Step 4: Simplify the Expression

Simplify the expression to find the work done: \[ W = 20 \times 100 \times \frac{\sqrt{3}}{2} = 2000 \times \frac{\sqrt{3}}{2} \] \[ W = 1000 \sqrt{3} \]

Key Concepts

Work formula

Work in physics is a measure of energy transfer that occurs when a force is applied to move an object over a distance. The general formula to calculate work (\(W\)) is: