A hyperboloid cooling tower is 400 feet wide at its base and is 350 feet wide at its narrowest point and 400 feet above the ground. What is the equation of the hyperboloid that models the shape of the tower?
The figure's base is the circle x^2+y^2 = 200^2 at z = -400
The equation of the hyperboloid is
x^2/175^2 + y^2/175^2 - z^2/c^2 = 1
and contains the point (200,0,-400)
so that makes it
x^2/175^2 + y^2/175^2 - z^2/(560 √(5/3))^2 = 1
To determine the equation of the hyperboloid, we need to use the standard equation for a hyperboloid of one sheet, which is:
(x^2/a^2) - (y^2/b^2) - (z^2/c^2) = 1
where a, b, and c are the parameters that determine the size and shape of the hyperboloid.
Given the dimensions of the hyperboloid cooling tower, we can find the values of a, b, and c.
At the base of the tower, the width is 400 feet, which corresponds to the value of a. At the narrowest point, the width is 350 feet, which corresponds to the value of b. The height of the tower is 400 feet, which corresponds to the value of c.
Therefore, the equation of the hyperboloid that models the shape of the tower is:
(x^2/400^2) - (y^2/350^2) - (z^2/400^2) = 1
To find the equation of the hyperboloid that models the shape of the tower, we need to know the equation for a hyperboloid of one sheet. The standard equation for a hyperboloid of one sheet can be written as:
(x^2 / a^2) + (y^2 / b^2) - (z^2 / c^2) = 1
Where 'a', 'b', and 'c' are constants that determine the shape and size of the hyperboloid.
Given the dimensions of the hyperboloid cooling tower, we can determine the values of 'a', 'b', and 'c' using the following information:
1. The tower is 400 feet wide at its base: In the x-direction, the width is 400 feet. This implies that a = 400/2 = 200.
2. The tower is 350 feet wide at its narrowest point: In the y-direction, the width is 350 feet. This implies that b = 350/2 = 175.
3. The tower is 400 feet above the ground: In the z-direction, the height is 400 feet. This implies that c = 400.
Substituting these values into the standard equation, we have:
(x^2 / 200^2) + (y^2 / 175^2) - (z^2 / 400^2) = 1
Thus, the equation of the hyperboloid that models the shape of the tower is:
(x^2 / 40000) + (y^2 / 30625) - (z^2 / 160000) = 1