Question

Sam is framing a closet under a stairway. the stairway is 13 ft 5 in.​ long, and its angle of elevation is 34 degrees°. find the depth of the closet to the nearest inch.

Answer 1

The depth of the closet is 90 inches.

Explanation:

Given : Sam is framing a closet under a stairway. The stairway is 13 ft 5 in.​ long, and its angle of elevation is 34 degrees°.

To find : The depth of the closet to the nearest inch?

Solution :

According to question,

Firstly we form a right angle triangle ABC, forming an image of a closet under a stairway.

Refer the attached figure below.

The stairway is 13 ft 5 in.​ long i.e, the hypotenuse of the stairways.

AC= 13 ft. 5 in.

Conversion of feet into inches,

1 feet = 12 inches

13 feet = 13 × 12=156 inches

Total AC=156+5=161 inches.

Its angle of elevation is 34 degrees°.

i.e,
`\theta=\angle ACB=34^\circ`

We have to find the dept of the closet i.e, AB

Applying trigonometry property of right angle triangle,

`\sin\theta=(Perpendicular)/(Hypotenuse)`

`\sin\theta=(AB)/(AC)`

`\sin(34)=(AB)/(161)`

`AB=\sin(34)* 161`

`AB=0.559* 161`

`AB=90.03in.`

Therefore, The depth of the closet is 85 inches.

Answer 2

90 inches.

Step-by-step explanation:

Please find the attachment.

We have been given that Sam is framing a closet under a stairway. The length of the stairway is 13 ft 5 inch and its angle of elevation is 34 degrees°.

Let us convert length of staircase in inches.

`13\text{ ft}+5\text{ inches}=((13* 12)+5)\text{ inches}=161\text{ inches}`

We can see that closet under staircase is forming a right triangle with hypotenuse of 13 ft and 5 inches. The depth of the closet is opposite of 34 degrees angle.

Since we know that sine relates the hypotenuse and opposite side of a given angle (in a right triangle), therefore we will use sine to find the depth of closet.

`sin(34)=\frac{\text{Depth}}{161}`

`\text{Depth}=161\cdot sin(34)`

`\text{Depth}=161\cdot 0.559192903471`

`\text{Depth}=90.030057458831 \approx 90`

Therefore, the depth of the closet to the nearest inch will be 90 inches.