Math / Trigonometry

QuestionOSHA safety regulations require that the base of a ladder be placed 1 ft from the wall for every 4 ft of ladder length. To the nearest tenth of a degree, find the angle that the ladder forms with the ground and the angle that it forms with the wall.

Studdy Solution

STEP 1

1. The ladder forms a right triangle with the wall and the ground.
2. The base of the ladder is placed 1 foot away from the wall for every 4 feet of ladder length.

STEP 2

1. Define the right triangle and its sides.
2. Use trigonometric functions to find the angle with the ground.
3. Use trigonometric functions to find the angle with the wall.

STEP 3

Define the right triangle: - Let $L$ be the length of the ladder. - The base of the triangle (distance from the wall) is $\frac{L}{4}$ . - The ladder itself is the hypotenuse.

STEP 4

Use trigonometric functions to find the angle $\theta$ that the ladder forms with the ground: - Use the cosine function, which relates the adjacent side (base) to the hypotenuse:
$\cos(\theta) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{\frac{L}{4}}{L} = \frac{1}{4}$
- Solve for $\theta$ :
$\theta = \cos^{-1}\left(\frac{1}{4}\right)$
- Calculate $\theta$ to the nearest tenth of a degree using a calculator:
$\theta \approx 75.5^\circ$

STEP 5

Use trigonometric functions to find the angle $\phi$ that the ladder forms with the wall: - Use the sine function, which relates the opposite side (base) to the hypotenuse:
$\sin(\phi) = \frac{\text{Base}}{\text{Hypotenuse}} = \frac{\frac{L}{4}}{L} = \frac{1}{4}$
- Solve for $\phi$ :
$\phi = \sin^{-1}\left(\frac{1}{4}\right)$
- Calculate $\phi$ to the nearest tenth of a degree using a calculator:
$\phi \approx 14.5^\circ$
The angle that the ladder forms with the ground is approximately $\boxed{75.5^\circ}$ and with the wall is approximately $\boxed{14.5^\circ}$ .

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms