Math / Geometry

QuestionConsult the figure. To find the length of the span of a oroposed ski lift from $A$ to $B$ , a surveyor measures the angle DAB to be $25^{\circ}$ and then walks off a distance of $L=1000$ feet to $C$ and measures the angle $A C B$ to be $15^{\circ}$ . What is the distance from $A$ to $B$ ?
The distance from $A$ to $B$ is approtmately $\square$ feet. (Do not round until the final answer. Then round to two decimal places as needed.)

Studdy Solution

STEP 1

1. We are given a triangle with points $A$ , $B$ , and $C$ .
2. The angle $\angle DAB = 25^\circ$ .
3. The angle $\angle ACB = 15^\circ$ .
4. The distance $AC = 1000$ feet.
5. We need to find the distance $AB$ .

STEP 2

1. Analyze the given information and determine the triangle type.
2. Use the Law of Sines to find the necessary angles and sides.
3. Calculate the distance from $A$ to $B$ using the derived values.

STEP 3

First, identify the triangle and the given angles. We have a triangle $\triangle ACB$ with $\angle ACB = 15^\circ$ and a line $AD$ such that $\angle DAB = 25^\circ$ . The distance from $A$ to $C$ is $1000$ feet.

STEP 4

To find the length $AB$ , we need to determine the angle $\angle CAB$ . Since $\angle DAB = 25^\circ$ and $\angle ACB = 15^\circ$ , the angle $\angle CAB$ is:
$\angle CAB = \angle DAB - \angle ACB = 25^\circ - 15^\circ = 10^\circ$

STEP 5

Now, use the Law of Sines in $\triangle ACB$ to find $AB$ . The Law of Sines states:
$\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle CAB)}$
Substitute the known values:
$\frac{AB}{\sin(15^\circ)} = \frac{1000}{\sin(10^\circ)}$

STEP 6

Solve for $AB$ :
$AB = \frac{1000 \cdot \sin(15^\circ)}{\sin(10^\circ)}$

STEP 7

Calculate the value using a calculator:
$AB \approx \frac{1000 \cdot 0.2588}{0.1736}$
$AB \approx \frac{258.8}{0.1736}$
$AB \approx 1490.09$
The distance from $A$ to $B$ is approximately $\boxed{1490.09}$ feet.

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Studdy Solution

STEP 1

1. We are given a triangle with points $A$ , $B$ , and $C$ .
2. The angle $\angle DAB = 25^\circ$ .
3. The angle $\angle ACB = 15^\circ$ .
4. The distance $AC = 1000$ feet.
5. We need to find the distance $AB$ .

STEP 2

1. Analyze the given information and determine the triangle type.
2. Use the Law of Sines to find the necessary angles and sides.
3. Calculate the distance from $A$ to $B$ using the derived values.

STEP 3

First, identify the triangle and the given angles. We have a triangle $\triangle ACB$ with $\angle ACB = 15^\circ$ and a line $AD$ such that $\angle DAB = 25^\circ$ . The distance from $A$ to $C$ is $1000$ feet.

STEP 4

To find the length $AB$ , we need to determine the angle $\angle CAB$ . Since $\angle DAB = 25^\circ$ and $\angle ACB = 15^\circ$ , the angle $\angle CAB$ is:
$\angle CAB = \angle DAB - \angle ACB = 25^\circ - 15^\circ = 10^\circ$

STEP 5

Now, use the Law of Sines in $\triangle ACB$ to find $AB$ . The Law of Sines states:
$\frac{AB}{\sin(\angle ACB)} = \frac{AC}{\sin(\angle CAB)}$
Substitute the known values:
$\frac{AB}{\sin(15^\circ)} = \frac{1000}{\sin(10^\circ)}$

STEP 6

Solve for $AB$ :
$AB = \frac{1000 \cdot \sin(15^\circ)}{\sin(10^\circ)}$

STEP 7

Calculate the value using a calculator:
$AB \approx \frac{1000 \cdot 0.2588}{0.1736}$
$AB \approx \frac{258.8}{0.1736}$
$AB \approx 1490.09$
The distance from $A$ to $B$ is approximately $\boxed{1490.09}$ feet.

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Studdy Solution

STEP 1

1. We are given a triangle with points A A A, B B B, and C C C.2. The angle ∠DAB=25∘ \angle DAB = 25^\circ ∠DAB=25∘.3. The angle ∠ACB=15∘ \angle ACB = 15^\circ ∠ACB=15∘.4. The distance AC=1000 AC = 1000 AC=1000 feet.5. We need to find the distance AB AB AB.

STEP 2

1. Analyze the given information and determine the triangle type.2. Use the Law of Sines to find the necessary angles and sides.3. Calculate the distance from A A A to B B B using the derived values.

STEP 3

First, identify the triangle and the given angles. We have a triangle △ACB \triangle ACB △ACB with ∠ACB=15∘ \angle ACB = 15^\circ ∠ACB=15∘ and a line AD AD AD such that ∠DAB=25∘ \angle DAB = 25^\circ ∠DAB=25∘. The distance from A A A to C C C is 1000 1000 1000 feet.

STEP 4

STEP 5

STEP 6

Solve for AB AB AB:AB=1000⋅sin⁡(15∘)sin⁡(10∘) AB = \frac{1000 \cdot \sin(15^\circ)}{\sin(10^\circ)} AB=sin(10∘)1000⋅sin(15∘)​

STEP 7

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1. We are given a triangle with points $A$ , $B$ , and $C$ .
2. The angle $\angle DAB = 25^\circ$ .
3. The angle $\angle ACB = 15^\circ$ .
4. The distance $AC = 1000$ feet.
5. We need to find the distance $AB$ .

1. Analyze the given information and determine the triangle type.
2. Use the Law of Sines to find the necessary angles and sides.
3. Calculate the distance from $A$ to $B$ using the derived values.

First, identify the triangle and the given angles. We have a triangle $\triangle ACB$ with $\angle ACB = 15^\circ$ and a line $AD$ such that $\angle DAB = 25^\circ$ . The distance from $A$ to $C$ is $1000$ feet.

Solve for $AB$ :
$AB = \frac{1000 \cdot \sin(15^\circ)}{\sin(10^\circ)}$