Math / Trigonometry

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An electrician leans an extension ladder against the outside wall of a house so that it reaches an electric box 22 feet up. The ladder makes an angle of $77^{\circ}$ with the ground. Find the length of the ladder. Round your answer to the nearest tenth of a foot if necessary. Answer Attempt 1 out of 2

Studdy Solution

STEP 1

1. The ladder forms a right triangle with the wall and the ground.
2. The angle between the ladder and the ground is $77^\circ$ .
3. The height the ladder reaches on the wall is $22$ feet.
4. We need to find the hypotenuse of the right triangle, which is the length of the ladder.

STEP 2

1. Recall the trigonometric relationship for right triangles.
2. Set up the equation using the sine function.
3. Solve for the length of the ladder.
4. Round the answer to the nearest tenth.

STEP 3

Recall the trigonometric relationship for right triangles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

STEP 4

Set up the equation using the sine function with the given angle and height:
$\sin(77^\circ) = \frac{22}{\text{Hypotenuse}}$

STEP 5

Solve for the length of the ladder (Hypotenuse):
$\text{Hypotenuse} = \frac{22}{\sin(77^\circ)}$
Calculate the sine of $77^\circ$ and then the hypotenuse:
$\sin(77^\circ) \approx 0.9744$
$\text{Hypotenuse} \approx \frac{22}{0.9744}$
$\text{Hypotenuse} \approx 22.58$

STEP 6

Round the answer to the nearest tenth:
The length of the ladder is approximately $\boxed{22.6}$ feet.

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Studdy Solution

STEP 1

1. The ladder forms a right triangle with the wall and the ground.
2. The angle between the ladder and the ground is $77^\circ$ .
3. The height the ladder reaches on the wall is $22$ feet.
4. We need to find the hypotenuse of the right triangle, which is the length of the ladder.

STEP 2

1. Recall the trigonometric relationship for right triangles.
2. Set up the equation using the sine function.
3. Solve for the length of the ladder.
4. Round the answer to the nearest tenth.

STEP 3

Recall the trigonometric relationship for right triangles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

STEP 4

Set up the equation using the sine function with the given angle and height:
$\sin(77^\circ) = \frac{22}{\text{Hypotenuse}}$

STEP 5

Solve for the length of the ladder (Hypotenuse):
$\text{Hypotenuse} = \frac{22}{\sin(77^\circ)}$
Calculate the sine of $77^\circ$ and then the hypotenuse:
$\sin(77^\circ) \approx 0.9744$
$\text{Hypotenuse} \approx \frac{22}{0.9744}$
$\text{Hypotenuse} \approx 22.58$

STEP 6

Round the answer to the nearest tenth:
The length of the ladder is approximately $\boxed{22.6}$ feet.

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Studdy Solution

STEP 1

1. The ladder forms a right triangle with the wall and the ground.2. The angle between the ladder and the ground is 77∘ 77^\circ 77∘.3. The height the ladder reaches on the wall is 22 22 22 feet.4. We need to find the hypotenuse of the right triangle, which is the length of the ladder.

STEP 2

1. Recall the trigonometric relationship for right triangles.2. Set up the equation using the sine function.3. Solve for the length of the ladder.4. Round the answer to the nearest tenth.

STEP 3

Recall the trigonometric relationship for right triangles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse:sin⁡(θ)=OppositeHypotenuse \sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}} sin(θ)=HypotenuseOpposite​

STEP 4

Set up the equation using the sine function with the given angle and height:sin⁡(77∘)=22Hypotenuse \sin(77^\circ) = \frac{22}{\text{Hypotenuse}} sin(77∘)=Hypotenuse22​

STEP 5

STEP 6

Round the answer to the nearest tenth:The length of the ladder is approximately 22.6 \boxed{22.6} 22.6​ feet.

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1. The ladder forms a right triangle with the wall and the ground.
2. The angle between the ladder and the ground is $77^\circ$ .
3. The height the ladder reaches on the wall is $22$ feet.
4. We need to find the hypotenuse of the right triangle, which is the length of the ladder.

1. Recall the trigonometric relationship for right triangles.
2. Set up the equation using the sine function.
3. Solve for the length of the ladder.
4. Round the answer to the nearest tenth.

Recall the trigonometric relationship for right triangles. The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse:
$\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

Set up the equation using the sine function with the given angle and height:
$\sin(77^\circ) = \frac{22}{\text{Hypotenuse}}$

Round the answer to the nearest tenth:
The length of the ladder is approximately $\boxed{22.6}$ feet.