Exemplo 6 de Algebra and Trigonometry, 12.2 The Hyperbola

Solving Applied Problems Involving Hyperbolas

Problema e solução em inglês, como no livro original.

The design layout of a cooling tower is shown in Figure 11. The tower stands 179.6 meters tall. The diameter of the top is 72 meters. At their closest, the sides of the tower are 60 meters apart.

Figure 11 Project design for a natural draft cooling tower

Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

Resolva no papel primeiro. Depois abra a solução um passo de cada vez e pare assim que conseguir terminar sozinho.

Solução
We are assuming the center of the tower is at the origin, so we can use the standard form of a horizontal hyperbola centered at the origin: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1,$ where the branches of the hyperbola form the sides of the cooling tower. We must find the values of $a^{2}$ and $b^{2}$ to complete the model.
First, we find $a^{2} .$ Recall that the length of the transverse axis of a hyperbola is $2 a .$ This length is represented by the distance where the sides are closest, which is given as $60$ meters. So, $2 a = 60.$ Therefore, $a = 30$ and $a^{2} = 900.$
To solve for $b^{2},$ we need to substitute for $x$ and $y$ in our equation using a known point. To do this, we can use the dimensions of the tower to find some point $(x, y)$ that lies on the hyperbola. We will use the top right corner of the tower to represent that point. Since the y-axis bisects the tower, our x-value can be represented by the radius of the top, or 36 meters. The y-value is represented by the distance from the origin to the top, which is given as 79.6 meters. Therefore,

$\begin{array}{l} \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 & \begin{matrix} \end{matrix} Standard form of horizontal hyperbola . \\ b^{2} = \frac{y^{2}}{\frac{x^{2}}{a^{2}} - 1} & \begin{matrix} \end{matrix} Isolate b^{2} \\ = \frac{{(79.6)}^{2}}{\frac{{(36)}^{2}}{900} - 1} & \begin{matrix} \end{matrix} Substitute for a^{2}, x, and y \\ \approx 14400.3636 & \begin{matrix} \end{matrix} Round to four decimal places \end{array}$
The sides of the tower can be modeled by the hyperbolic equation

$\frac{x^{2}}{900} - \frac{y^{2}}{14400.3636} = 1, or \frac{x^{2}}{30^{2}} - \frac{y^{2}}{{120.0015}^{2}} = 1$

Sua vez

O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

A design for a cooling tower project is shown in Figure 12. Find the equation of the hyperbola that models the sides of the cooling tower. Assume that the center of the hyperbola—indicated by the intersection of dashed perpendicular lines in the figure—is the origin of the coordinate plane. Round final values to four decimal places.

Project design for a natural draft cooling tower. The overall height is 167.082 meters. The diameter at the top is 60 meters, and at their closest, 79.6 meters from the top, the sides are 60 meters apart.

Figure 12

Como foi?

Sua vez

O Try It do livro logo depois deste exemplo: mesma ideia, outros números. Só a resposta é dada.

Figure 12

Solução

Sua vez

Solução

Sua vez