(Modeling) Grade Resistance Solve each problem. See Example 3. Find the grade resistance, to the nearest ten pounds, for a 2400-lb car traveling on a -2.4° downhill grade.

Verified step by step guidance

1. Understand that grade resistance is the component of the car's weight acting along the slope due to gravity. It can be found using the formula: \(\text{Grade Resistance} = W \times \sin(\theta)\), where \(W\) is the weight of the car and \(\theta\) is the angle of the slope.
2. Identify the given values: the weight of the car \(W = 2400\) pounds, and the slope angle \(\theta = -2.4^\circ\). The negative sign indicates a downhill grade.
3. Calculate the sine of the angle \(\theta\). Since the angle is negative, \(\sin(-2.4^\circ)\) will be negative, reflecting the downhill direction.
4. Multiply the weight \(W\) by \(\sin(\theta)\) to find the grade resistance: \(2400 \times \sin(-2.4^\circ)\).
5. Interpret the result: the grade resistance will be negative, indicating it acts in the direction of motion (downhill). Round the absolute value of the result to the nearest ten pounds as requested.

Key Concepts

Grade Resistance

Grade resistance is the component of gravitational force acting against the motion of a vehicle on an inclined surface. It depends on the vehicle's weight and the slope angle, calculated as the weight multiplied by the sine of the grade angle. This force affects how much effort is needed to move the vehicle uphill or downhill.

Trigonometric Functions and Angles

Understanding sine and its relation to angles is essential for calculating forces on slopes. The sine of the grade angle gives the ratio of the vertical height change to the hypotenuse (road length), allowing conversion of the slope angle into a force component. Negative angles indicate downhill grades, affecting the direction of the force.

Force Calculation and Units

Calculating grade resistance requires multiplying the vehicle's weight (in pounds) by the sine of the grade angle to find the force in pounds. Proper rounding, as requested to the nearest ten pounds, ensures practical and usable results. Understanding units and rounding conventions is crucial for accurate and meaningful answers.