Chapter 4, Problem 56CR

(a)

To determine

To form: the trigonometric function for the unknown value.

Answer to Problem 56CR

Required trigonometric equation is $\underset{\_}{x=152\sin {30}^{\circ }}$ where $x$ is height of the platform.

Explanation of Solution

Given information:

1) Length of the escalator is $152ft$ .

2) Angle that escalator makes with the ground is ${30}^{\circ }$ .

Formula used:

  $\sin \theta =\frac{\text{side opposite}\text{to}\theta }{\text{hypotenuse}}$

Calculation:

Consider the following figure

  

Length of the escalator is represented by AC $=152\text{feet}$

Assume height of the platform that is AB $=x\text{feet}$

Ground is represented by BC.

  $\sin {30}^{\circ }=\frac{\text{AB}}{\text{AC}}$  [using $\sin \theta =\frac{\text{side opposite}\text{to}\theta }{\text{hypotenuse}}$ ]

  $\sin {30}^{\circ }=\frac{x}{\text{152}}$

  $152\sin {30}^{\circ }=x$

  $x=152\sin {30}^{\circ }$

Hence required trigonometric equation is $\underset{\_}{x=152\sin {30}^{\circ }}$ where $x$ is height of the platform.

(b)

To determine

To calculate: the height of the platform above the ground.

Answer to Problem 56CR

Height of the platform above the ground is $76\text{ft}$ .

Explanation of Solution

Given information:

1. Length of the escalator is $152ft$ .
2. Angle that escalator makes with the ground is ${30}^{\circ }$ .
3. Evaluated equation for height of the platform is $x=152\sin {30}^{\circ }$ .

Formula used:

  $\sin \theta =\frac{\text{side opposite}\text{to}\theta }{\text{hypotenuse}}$

Calculation:

Consider the following figure

  

Length of the escalator is represented by AC $=152\text{feet}$

Assume height of the platform that is AB $=x\text{feet}$

Ground is represented by BC.

  $\sin {30}^{\circ }=\frac{\text{AB}}{\text{AC}}$   [using $\sin \theta =\frac{\text{side opposite}\text{to}\theta }{\text{hypotenuse}}$ ]

  $\sin {30}^{\circ }=\frac{x}{\text{152}}$

  $152\sin {30}^{\circ }=x$

  $x=152\sin {30}^{\circ }$

  $x=152\times \frac{1}{2}=76\text{ft}$

Hence height of the platform above the ground is $\underset{\_}{76\text{ft}}$ .