Final answer:
To calculate the force that the rope must exert on the cart to keep it from rolling down the inclined ramp, one must find the component of the cart's weight parallel to the incline and then use the angle of the rope to the horizontal to resolve the force needed from the rope to counteract this component.
Step-by-step explanation:
The given scenario involves a cart on an incline and requires the use of physics to calculate the necessary force from a rope to prevent the cart from sliding down. To solve this problem, we first need to consider the components of the weight of the cart along the incline. We can find the component of the cart's weight parallel to the incline, which will be w sin(15°), where w is the weight of the cart (60 lb).
Secondly, we factor in the angle of the rope to the horizontal (60°). The force exerted by the rope has to counteract the component of the cart's weight that is trying to pull it down the incline. Thus, we need to resolve the force of the rope into components as well and focus on the component parallel to the incline that opposes the cart's downward motion. The force of the rope perpendicular to the incline contributes to the normal force and does not affect the cart's tendency to slide down. Therefore, the force of the rope that acts to keep the cart stationary is equal to the component of the cart's weight that is parallel to the incline.
Given these considerations, the force the rope must exert, F, can be calculated using the formula:
F = w sin(15°) / sin(60°)
After calculating F, round the result to one decimal place to obtain the force in pounds. This force is the tension in the rope required to keep the cart stationary on the inclined ramp.