Math / Geometry

QuestionApplications Draw well labeled diagrams if one is not provided. Show ALL WORK and FORMULAS! /a) Mr. Jonsen is trying to hit his golf ball between two trees. He estimates the distances shown. Within what angle must Mr. Jensen make his shot, in order to pass between the trees? Round to the nearest tenth of a degree.

Studdy Solution

STEP 1

What is this asking? We need to find the angle formed at the golfer's position, between the lines of sight to the two trees. Watch out! Remember, we're looking for the angle at the golfer's position, not any other angle in the triangle!

STEP 2

1. Set up the problem
2. Calculate the angle

STEP 3

Imagine a triangle with the golfer at one vertex and the two trees at the other two vertices.
The distance between the trees is the side opposite to the golfer's angle.

STEP 4

We know the distance between the trees ( $20$ meters) and the distances from the golfer to each tree ( $78$ meters and $80$ meters).

STEP 5

Since we know three sides of the triangle, we can use the Law of Cosines to find the angle.

STEP 6

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ , where $c$ is the side opposite angle $C$ , and $a$ and $b$ are the other two sides.

STEP 7

In our case, let $a = 78$ , $b = 80$ , and $c = 20$ .
We want to find angle $C$ .
So, we have: $20^2 = 78^2 + 80^2 - 2 \cdot 78 \cdot 80 \cdot \cos(C)$

STEP 8

$400 = 6084 + 6400 - 12480 \cdot \cos(C)$ $400 = 12484 - 12480 \cdot \cos(C)$

STEP 9

To isolate the cosine term, we subtract $12484$ from both sides: $400 - 12484 = -12480 \cdot \cos(C)$ $-12084 = -12480 \cdot \cos(C)$

STEP 10

Now, divide both sides by $-12480$ to solve for $\cos(C)$ : $\cos(C) = \frac{-12084}{-12480}$ $\cos(C) \approx 0.968269230769$

STEP 11

Finally, take the inverse cosine (arccos) of both sides to find angle $C$ : $C = \arccos(0.968269230769)$ $C \approx 14.6^\circ$

STEP 12

Mr. Jensen must hit his golf ball within an angle of approximately $14.6$ degrees to pass between the trees.

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Studdy Solution

STEP 1

What is this asking? We need to find the angle formed at the golfer's position, between the lines of sight to the two trees. Watch out! Remember, we're looking for the angle at the golfer's position, not any other angle in the triangle!

STEP 2

1. Set up the problem
2. Calculate the angle

STEP 3

Imagine a triangle with the golfer at one vertex and the two trees at the other two vertices.
The distance between the trees is the side opposite to the golfer's angle.

STEP 4

We know the distance between the trees ( $20$ meters) and the distances from the golfer to each tree ( $78$ meters and $80$ meters).

STEP 5

Since we know three sides of the triangle, we can use the Law of Cosines to find the angle.

STEP 6

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ , where $c$ is the side opposite angle $C$ , and $a$ and $b$ are the other two sides.

STEP 7

In our case, let $a = 78$ , $b = 80$ , and $c = 20$ .
We want to find angle $C$ .
So, we have: $20^2 = 78^2 + 80^2 - 2 \cdot 78 \cdot 80 \cdot \cos(C)$

STEP 8

$400 = 6084 + 6400 - 12480 \cdot \cos(C)$ $400 = 12484 - 12480 \cdot \cos(C)$

STEP 9

To isolate the cosine term, we subtract $12484$ from both sides: $400 - 12484 = -12480 \cdot \cos(C)$ $-12084 = -12480 \cdot \cos(C)$

STEP 10

Now, divide both sides by $-12480$ to solve for $\cos(C)$ : $\cos(C) = \frac{-12084}{-12480}$ $\cos(C) \approx 0.968269230769$

STEP 11

Finally, take the inverse cosine (arccos) of both sides to find angle $C$ : $C = \arccos(0.968269230769)$ $C \approx 14.6^\circ$

STEP 12

Mr. Jensen must hit his golf ball within an angle of approximately $14.6$ degrees to pass between the trees.

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Studdy Solution

STEP 1

What is this asking? We need to find the angle formed at the golfer's position, between the lines of sight to the two trees. Watch out! Remember, we're looking for the angle at the golfer's position, not any other angle in the triangle!

STEP 2

1. Set up the problem2. Calculate the angle

STEP 3

Imagine a triangle with the golfer at one vertex and the two trees at the other two vertices.The distance between the trees is the side opposite to the golfer's angle.

STEP 4

We know the distance between the trees (202020 meters) and the distances from the golfer to each tree (787878 meters and 808080 meters).

STEP 5

Since we know three sides of the triangle, we can use the Law of Cosines to find the angle.

STEP 6

The Law of Cosines states: c2=a2+b2−2⋅a⋅b⋅cos⁡(C)c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)c2=a2+b2−2⋅a⋅b⋅cos(C), where ccc is the side opposite angle CCC, and aaa and bbb are the other two sides.

STEP 7

In our case, let a=78a = 78a=78, b=80b = 80b=80, and c=20c = 20c=20.We want to find angle CCC.So, we have: 202=782+802−2⋅78⋅80⋅cos⁡(C)20^2 = 78^2 + 80^2 - 2 \cdot 78 \cdot 80 \cdot \cos(C)202=782+802−2⋅78⋅80⋅cos(C)

STEP 8

400=6084+6400−12480⋅cos⁡(C)400 = 6084 + 6400 - 12480 \cdot \cos(C)400=6084+6400−12480⋅cos(C) 400=12484−12480⋅cos⁡(C)400 = 12484 - 12480 \cdot \cos(C)400=12484−12480⋅cos(C)

STEP 9

To isolate the cosine term, we subtract 124841248412484 from both sides: 400−12484=−12480⋅cos⁡(C)400 - 12484 = -12480 \cdot \cos(C)400−12484=−12480⋅cos(C) −12084=−12480⋅cos⁡(C)-12084 = -12480 \cdot \cos(C)−12084=−12480⋅cos(C)

STEP 10

Now, divide both sides by −12480-12480−12480 to solve for cos⁡(C)\cos(C)cos(C): cos⁡(C)=−12084−12480\cos(C) = \frac{-12084}{-12480}cos(C)=−12480−12084​ cos⁡(C)≈0.968269230769\cos(C) \approx 0.968269230769cos(C)≈0.968269230769

STEP 11

Finally, take the inverse cosine (arccos) of both sides to find angle CCC: C=arccos⁡(0.968269230769)C = \arccos(0.968269230769)C=arccos(0.968269230769) C≈14.6∘C \approx 14.6^\circC≈14.6∘

STEP 12

Mr. Jensen must hit his golf ball within an angle of approximately **14.614.614.6 degrees** to pass between the trees.

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1. Set up the problem
2. Calculate the angle

Imagine a triangle with the golfer at one vertex and the two trees at the other two vertices.
The distance between the trees is the side opposite to the golfer's angle.

We know the distance between the trees ( $20$ meters) and the distances from the golfer to each tree ( $78$ meters and $80$ meters).

The Law of Cosines states: $c^2 = a^2 + b^2 - 2 \cdot a \cdot b \cdot \cos(C)$ , where $c$ is the side opposite angle $C$ , and $a$ and $b$ are the other two sides.

In our case, let $a = 78$ , $b = 80$ , and $c = 20$ .
We want to find angle $C$ .
So, we have: $20^2 = 78^2 + 80^2 - 2 \cdot 78 \cdot 80 \cdot \cos(C)$

$400 = 6084 + 6400 - 12480 \cdot \cos(C)$ $400 = 12484 - 12480 \cdot \cos(C)$

To isolate the cosine term, we subtract $12484$ from both sides: $400 - 12484 = -12480 \cdot \cos(C)$ $-12084 = -12480 \cdot \cos(C)$

Now, divide both sides by $-12480$ to solve for $\cos(C)$ : $\cos(C) = \frac{-12084}{-12480}$ $\cos(C) \approx 0.968269230769$

Finally, take the inverse cosine (arccos) of both sides to find angle $C$ : $C = \arccos(0.968269230769)$ $C \approx 14.6^\circ$

Mr. Jensen must hit his golf ball within an angle of approximately $14.6$ degrees to pass between the trees.