A cowboy is practicing his lassoing technique by lassoing the fence posts at his ranch. Consider that is lariat forms a perecly elliptical shape centered at the origin while he twirls it. If the fence posts are 2 inches in circumference and the cowboy would lik to add 5 inches extra "cushion" to the minimum dimension in setting his lariat, what should the length of the minor axis be? If the cowboy likes to set his lariat so that the major axis is twice the minor axi, deermine the equation of the ellipse which models the twieling path of his lariat.

To determine the length of the minor axis, we first need to find the minimum circumference of the lariat's elliptical shape.

Given that the fence posts are 2 inches in circumference, we can calculate the minimum circumference of the lariat by adding the extra 5 inches cushion:

Minimum circumference of the lariat = 2 inches (fence post circumference) + 5 inches (extra cushion) = 7 inches

Since an ellipse has two axes, major and minor, we can consider the lengths of these axes as 2a and 2b, respectively.

From the given information, we know that the major axis is twice as long as the minor axis. Therefore, we have:

2a = 2 * 2b
2a = 4b
a = 2b

Now, let's find the equation of the ellipse that models the twirling path of his lariat.

The equation of an ellipse with the center at the origin (0,0) is:

(x^2 / a^2) + (y^2 / b^2) = 1

Since a = 2b, we can substitute this into the equation:

(x^2 / (2b)^2) + (y^2 / b^2) = 1
(x^2 / 4b^2) + (y^2 / b^2) = 1

Now let's find the value of b, the length of the minor axis:

The circumference of an ellipse is given by the formula:

Circumference = π * (3(a+b) - √((3a+b)(a+3b)))

For our lariat, the minimum circumference is 7 inches. Plugging this into the formula, we have:

7 = π * (3(2b+b) - √((3(2b)+b)(2b+3b)))

7 = π * (3(3b) - √((9b)(5b)))
7 = π * (9b - √(45b^2))
7 = π * (9b - 3√5 * b)

Now, solve for b:

7 = 9πb - 3π√5 * b
7 = (9π - 3π√5) * b
b = 7 / (9π - 3π√5)

Now that we have the value of b, we can substitute it back into the equation of the ellipse:

(x^2 / 4(7 / (9π - 3π√5))^2) + (y^2 / (7 / (9π - 3π√5))^2) = 1

Simplifying further is challenging without specific numerical values for π (pi) and √5. However, this is the general equation of the ellipse that models the twirling path of the cowboy's lariat, with the length of the minor axis determined by the given conditions.

To find the length of the minor axis of the elliptical shape formed by the cowboy's lariat, we'll start by considering the circumference of the fence post.

The circumference of a circle is given by the formula: C = 2πr, where C is the circumference and r is the radius. In this case, the circumference of the fence post is given as 2 inches, so 2 = 2πr. Solving for r gives us r = 1/π inches.

Now, let's add the "cushion" to the minimum dimension. The cushion is 5 inches, so the minimum radius of the ellipse would be r + 5 (in inches). In terms of our previous calculation, this would be (1/π) + 5 inches.

For an ellipse, the major axis (2a) is twice the length of the minor axis (2b). So, 2a = 2 * 2b. Simplifying this equation allows us to find the length of the minor axis (b).

Let's solve for b:
2a = 4b
2((1/π) + 5) = 4b
2/π + 10 = 4b
b = (2/π + 10)/4

Now that we have the length of the minor axis (b), we can find the equation of the ellipse.

The standard equation for an ellipse centered at the origin is: x^2/a^2 + y^2/b^2 = 1.

We already know the length of the minor axis (b). As for the major axis, it is twice the length of the minor axis, so the length of the major axis would be 2 * b.

The equation of the ellipse centered at the origin becomes:
x^2/(2b)^2 + y^2/b^2 = 1.

Simplifying, we have:
x^2/(4b^2) + y^2/b^2 = 1.

Therefore, the equation of the ellipse that models the twirling path of the cowboy's lariat is:
x^2/(4((2/π + 10)/4)^2) + y^2/((2/π + 10)/4)^2 = 1.