Exemplo 19 de Algebra and Trigonometry, 9.5 Solving Trigonometric Equations

Using the Pythagorean Theorem to Model an Abstract Problem

Problema e solução em inglês, como no livro original.

OSHA safety regulations require that the base of a ladder be placed 1 foot from the wall for every 4 feet of ladder length. Find the angle that a ladder of any length forms with the ground and the height at which the ladder touches the wall.

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

Solução
For any length of ladder, the base needs to be a distance from the wall equal to one fourth of the ladder’s length. Equivalently, if the base of the ladder is “a” feet from the wall, the length of the ladder will be 4a feet. See Figure 5.

Figure 5

The side adjacent to $θ$ is a and the hypotenuse is $4 a .$ Thus,

$\begin{matrix} \cos θ & = & \frac{a}{4 a} = \frac{1}{4} \\ \cos^{−1} (\frac{1}{4}) & \approx & 75.5° \end{matrix}$
The elevation of the ladder forms an angle of $75.5°$ with the ground. The height at which the ladder touches the wall can be found using the Pythagorean Theorem:
$\begin{matrix} a^{2} + b^{2} & = & {(4 a)}^{2} \\ b^{2} & = & {(4 a)}^{2} - a^{2} \\ b^{2} & = & 16 a^{2} - a^{2} \\ b^{2} & = & 15 a^{2} \\ b & = & a \sqrt{15} \end{matrix}$
Thus, the ladder touches the wall at $a \sqrt{15}$ feet from the ground.

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Solução

Solução