Exemplo 18 de Algebra and Trigonometry, 9.5 Solving Trigonometric Equations

Using the Pythagorean Theorem to Model an Equation

Problema e solução em inglês, como no livro original.

Use the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.

One of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is 69.5 meters above the ground, and the second anchor on the ground is 23 meters from the base of the Ferris wheel. Approximately how long is the cable, and what is the angle of elevation (from ground up to the center of the Ferris wheel)? See Figure 4.

Basic diagram of a ferris wheel (circle) and its support cables (form a right triangle). One cable runs from the center of the circle to the ground (outside the circle), is perpendicular to the ground, and has length 69.5. Another cable of unknown length (the hypotenuse) runs from the center of the circle to the ground 23 feet away from the other cable at an angle of theta degrees with the ground. So, in closing, there is a right triangle with base 23, height 69.5, hypotenuse unknown, and angle between base and hypotenuse of theta degrees.

Figure 4

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

Solução
Using the information given, we can draw a right triangle. We can find the length of the cable with the Pythagorean Theorem.

$\begin{matrix} a^{2} + b^{2} & = & c^{2} \\ {(23)}^{2} + {(69.5)}^{2} & \approx & 5359 \\ \sqrt{5359} & \approx & 73.2 m \end{matrix}$
The angle of elevation is $θ,$ formed by the second anchor on the ground and the cable reaching to the center of the wheel. We can use the tangent function to find its measure. Round to two decimal places.
$\begin{matrix} \tan θ & = & \frac{69.5}{23} \\ \tan^{−1} (\frac{69.5}{23}) & \approx & 1.2522 \\ \approx & 71.69° \end{matrix}$
The angle of elevation is approximately $71.7°,$ and the length of the cable is 73.2 meters.

Como foi?

Solução

Solução