A lantern suspended from a verandah roof by a 50 cm chain is blown by the wind so that it hangs at an angle θ to the vertical for the duration of the wind gust. If the wind blows from the east and exerts a constant force of 20 N, determine the tension, T , in the chain and the angle θ

This problem can be solved using the principles of trigonometry and physics. Here are the steps:

1. First, we need to understand that the tension in the chain, T, is the resultant force of the weight of the lantern and the force of the wind. The weight of the lantern acts vertically downwards and the wind force acts horizontally.

2. The weight of the lantern can be calculated using the formula: Weight = mass * gravity. However, the mass of the lantern is not given in the problem. Therefore, we cannot calculate the exact value of the weight or the tension in the chain. But we can express the weight in terms of its mass (m) and gravity (g). So, the weight is m*g.

3. The tension in the chain, T, can be calculated using the Pythagorean theorem because it is the hypotenuse of the right triangle formed by the weight and the wind force. So, T = sqrt((wind force)^2 + (weight)^2) = sqrt((20N)^2 + (m*g)^2).

4. The angle θ can be calculated using the tangent function, which is the ratio of the opposite side (wind force) to the adjacent side (weight) in a right triangle. So, tan(θ) = wind force / weight = 20N / (mg). Therefore, θ = arctan(20N / (mg)).

5. To get the exact values of T and θ, we need to know the mass of the lantern and the acceleration due to gravity. If we assume the standard gravity g = 9.8 m/s^2, we can substitute it into the formulas. But without the mass, we can't get the numerical answers.

6. In conclusion, the tension in the chain and the angle at which the lantern hangs depend on the mass of the lantern. With the given information, we can only express them in terms of the mass.