Given an elliptical ranch with dimensions 2*R units horizontally and 1 unit vertically, a goat is tied to the fence inside the ranch using a leash of length r units. The angle formed between the point of tying and the horizontal axis is θ. Determine the area of grass the goat can graze.Given :R- half of the horizontal dimension of the ellipseθ - the angle formed by the point of tying and the horizontal axis.r - length of the leash
Question
Given an elliptical ranch with dimensions 2*R units horizontally and 1 unit vertically, a goat is tied to the fence inside the ranch using a leash of length r units. The angle formed between the point of tying and the horizontal axis is θ. Determine the area of grass the goat can graze.Given :R- half of the horizontal dimension of the ellipseθ - the angle formed by the point of tying and the horizontal axis.r - length of the leash
Solution
The area of grass the goat can graze can be determined by calculating the area of the sector of the ellipse that the goat can reach.
Step 1: Determine the radius of the sector The radius of the sector is the length of the leash, which is given as r units.
Step 2: Determine the angle of the sector The angle of the sector is given as θ.
Step 3: Calculate the area of the sector The area of a sector of an ellipse is given by the formula (1/2)abθ, where a and b are the semi-major and semi-minor axes of the ellipse, and θ is the angle of the sector in radians.
In this case, a = R (half of the horizontal dimension of the ellipse), b = 1 (the vertical dimension of the ellipse), and θ is given.
So, the area of the sector is (1/2)R1θ = (1/2)Rθ.
Step 4: Adjust for the leash length However, the goat can only graze within a circle of radius r (the length of the leash). So, we need to adjust the area of the sector to account for this.
The area of a circle is given by the formula πr^2. So, the area of the circle that the goat can graze is πr^2.
Step 5: Determine the area the goat can graze The area of grass the goat can graze is the smaller of the area of the sector and the area of the circle.
So, the area the goat can graze is min((1/2)Rθ, πr^2).
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