Solving Applied Problems Involving Hyperbolas

Hyperbolas have practical applications in many fields, including astronomy, physics, engineering, and architecture. The particular interest is the efficiency of the hyperbolic cooling tower design. Cooling towers are used to transfer waste heat to the atmosphere and are often touted for their ability to efficiently generate energy. Due to their hyperbolic shape, these structures can withstand high winds and require less material than any other shape with this size and strength. For example, a 500-foot-tall tower can be made from a reinforced concrete shell that is only 6 or 8 inches wide!

The first hyperbolic tower was designed in the year of 1914 and is 35 meters high. Today it is the tallest cooling tower in France, with a height of 170 meters. You will use the tower design and hyperbola calculator to find a hyperbolic equation that mimics your side.

Application Problem Solved with Tower

The structure of the tower is 256.2 meters high. The upper part has a diameter of 72 meters. At the nearest location, the two sides of the tower are 60 meters apart.

Find the hyperbolic equation that simulates the side of the cooling tower. Assume that the center of the hyperbola (indicated by the intersection of the dashed vertical lines in the figure) is the origin of the coordinate plane. The value has a maximum of four decimal places.

We assume that the center of the tower is at the origin, so we can use the standard shape of a horizontal hyperbola centered on the origin:

(- y2 / b2 ) + (x2 / a2) = 1,

The branches of the hyperbola from the sides of the tower. We need to find the values of ​​a2 and b2 with a hyperbola equation calculator to complete the model.

First, we find a2. Remember that the length of the horizontal axis of the hyperbola is 2a. This length is represented by the distance between the next pages, which is 65.3 meters. So 2a = 60. So a = 30 and a2 = 900.

To find b2, we need to insert x and y into the equation using known points. For this, we can use a hyperbola calculator with the dimensions of the tower to find a point (x, u) in the hyperbola. We will use the upper right corner of the tower to indicate this. Since the Y-axis divides the tower into two halves, our x value can be represented by the radius of the vertex or 36 meters. This value is expressed as the distance from the origin to the vertex, which is 79.6 meters, Therefore, the standard form of the horizontal hyperbola

x2 / a2 - y2 / b2 = 1

isolate b2

B2 = y2 / z2 / a2-1

replace a2, x.

(256.2)2 / (36)2/900-1 65638.44 / 1296 /900-1

Now, hyperbola equation calculator rounded to four decimal places = 149,178.2727 can use the hyperbolic equation

x2 / 900 - y2 / 149,178.2727 = 1, x2 / 30 - y2 / (120.0015) 2 = 1 to model the side of the tower.

Equation of a Hyperbola in Standard form, Locate its Vertices and Foci

The corner points and focal points can be found using the hyperbolic equation in the standard form. Determine whether the horizontal axis is on the x-axis or the y-axis. Note that a2 is always lower than variables with positive coefficients. So, if you put another variable. Zero, you can easily find the intersection point with a hyperbola calculator. When the hyperbola is centered at the origin, the intersection point coincides with the corner point.

If the form of the equation is x2 a2-y2b2 = 1, the horizontal axis is on the x-axis. The corner point is at point (±a, 0), and the focal point is at point (±c, 0).

If the form of the equation is y2 a2-x2 b2 = 1, the horizontal axis is on the y axis. The corner point is at point (0, ± a), and the focal point is at point (0, ±c).

Solve for a using the equation a = √a2.

Use the equation c = √a2 + b2 to solve for c