Math / Geometry

Question47. To estimate the width of an archaeological mound, archaeologists place two stakes on opposite ends of the widest point. See Eigure 8.51 . They set a third stake 82 feet from one stake and 97 feet from the other stake. The angle formed is $125^{\circ}$ . Find the width of the mound.
Figure 8.51

Studdy Solution

STEP 1

1. We are given a triangle formed by the three stakes.
2. One angle in the triangle is $125^\circ$ .
3. The distances from the third stake to the other two stakes are 82 feet and 97 feet.
4. We need to find the width of the mound, which is the distance between the two stakes on opposite ends of the widest point.

STEP 2

1. Identify the triangle and label the sides and angles.
2. Use the Law of Cosines to find the width of the mound.

STEP 3

Identify the triangle formed by the three stakes. Let's label the stakes as points $A$ , $B$ , and $C$ . Assume: - $A$ and $B$ are the stakes on opposite ends of the mound. - $C$ is the third stake. - The angle at $C$ is $125^\circ$ . - The distance $AC = 82$ feet. - The distance $BC = 97$ feet. - The width of the mound is the distance $AB$ .

STEP 4

Use the Law of Cosines to find the width $AB$ . The Law of Cosines states:
$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$
where $c$ is the side opposite angle $C$ , and $a$ and $b$ are the other two sides. In this case: - $a = 82$ feet - $b = 97$ feet - $C = 125^\circ$

STEP 5

Substitute the known values into the Law of Cosines formula:
$AB^2 = 82^2 + 97^2 - 2 \times 82 \times 97 \times \cos(125^\circ)$

STEP 6

Calculate each term:
1. $82^2 = 6724$
2. $97^2 = 9409$
3. $2 \times 82 \times 97 = 15908$
4. $\cos(125^\circ) \approx -0.5736$
Substitute these values back into the equation:
$AB^2 = 6724 + 9409 - 15908 \times (-0.5736)$

STEP 7

Calculate the expression:
$AB^2 = 6724 + 9409 + 9124.3488$ $AB^2 = 25257.3488$
Take the square root to find $AB$ :
$AB = \sqrt{25257.3488}$ $AB \approx 158.92$
The width of the mound is approximately $158.92$ feet.

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Studdy Solution

STEP 1

STEP 2

1. Identify the triangle and label the sides and angles.
2. Use the Law of Cosines to find the width of the mound.

STEP 3

STEP 4

Use the Law of Cosines to find the width $AB$ . The Law of Cosines states:
$c^2 = a^2 + b^2 - 2ab \cdot \cos(C)$
where $c$ is the side opposite angle $C$ , and $a$ and $b$ are the other two sides. In this case: - $a = 82$ feet - $b = 97$ feet - $C = 125^\circ$

STEP 5

Substitute the known values into the Law of Cosines formula:
$AB^2 = 82^2 + 97^2 - 2 \times 82 \times 97 \times \cos(125^\circ)$

STEP 6

Calculate each term:
1. $82^2 = 6724$
2. $97^2 = 9409$
3. $2 \times 82 \times 97 = 15908$
4. $\cos(125^\circ) \approx -0.5736$
Substitute these values back into the equation:
$AB^2 = 6724 + 9409 - 15908 \times (-0.5736)$

STEP 7

Calculate the expression:
$AB^2 = 6724 + 9409 + 9124.3488$ $AB^2 = 25257.3488$
Take the square root to find $AB$ :
$AB = \sqrt{25257.3488}$ $AB \approx 158.92$
The width of the mound is approximately $158.92$ feet.

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Studdy Solution

STEP 1

STEP 2

1. Identify the triangle and label the sides and angles.2. Use the Law of Cosines to find the width of the mound.

STEP 3

STEP 4

STEP 5

Substitute the known values into the Law of Cosines formula:AB2=822+972−2×82×97×cos⁡(125∘) AB^2 = 82^2 + 97^2 - 2 \times 82 \times 97 \times \cos(125^\circ) AB2=822+972−2×82×97×cos(125∘)

STEP 6

STEP 7

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1. Identify the triangle and label the sides and angles.
2. Use the Law of Cosines to find the width of the mound.

Substitute the known values into the Law of Cosines formula:
$AB^2 = 82^2 + 97^2 - 2 \times 82 \times 97 \times \cos(125^\circ)$