maker is designing a mainsail for a newly constructed sailboat. The mainsail foot is 13 feet long, the luff is 10 feet long, and the luff makes a 52° angle at the head with the leach. Head Leech Luff : Determine the length of the leach. 3: Find the area of the mainsail. Foot
maker is designing a mainsail for a newly constructed sailboat. The mainsail foot is 13 feet long, the luff is 10 feet long, and the luff makes a 52° angle at the head with the leach. Head Leech Luff : Determine the length of the leach. 3: Find the area of the mainsail. Foot
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter9: Real Numbers And Right Triangles
Section9.3: The Pythagorean Theorem
Problem 20E
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![### Designing a Mainsail for a Sailboat
A sailmaker is designing a mainsail for a newly constructed sailboat. The mainsail foot is 13 feet long, the luff is 10 feet long, and the luff makes a 52° angle at the head with the leech.
#### Diagram Description
The diagram depicts a triangular mainsail on a sailboat. The three sides of the triangular sail are labeled as follows:
- **Head**: The top vertex of the triangle where the luff and leech meet.
- **Luff**: The side of the triangle closest to the mast (left side), 10 feet long.
- **Leech**: The side opposite to the mast (right side).
- **Foot**: The bottom side of the triangle, 13 feet long.
A 52° angle is shown between the luff and the leech at the head of the sail.
### Calculations
#### Part A: Determine the length of the leech (c).
To determine the length of the leech, we can use the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
where
\(a = 10 \) feet (luff),
\(b = 13 \) feet (foot), and
\(C = 52° \).
By substituting these values into the formula:
\[ c^2 = 10^2 + 13^2 - 2(10)(13) \cos(52°) \]
\[ c^2 = 100 + 169 - 2(10)(13) \cos(52°) \]
\[ c^2 = 269 - 260 \cos(52°) \]
Calculate the value of \( \cos(52°) \):
\[ \cos(52°) \approx 0.6157 \]
Substituting this back into the equation:
\[ c^2 = 269 - 260 (0.6157) \]
\[ c^2 = 269 - 160.082 \]
\[ c^2 = 108.918 \]
\[ c \approx \sqrt{108.918} \]
\[ c \approx 10.43 \text{ feet} \]
So, the leech is approximately 10.43 feet long.
#### Part B: Find the area of the mains](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3e08c0ed-2478-4250-8ecc-80933702ca06%2F2a19769e-951d-4785-95e7-48f9eea09bf4%2Fjuyoqjsn_processed.png&w=3840&q=75)
Transcribed Image Text:### Designing a Mainsail for a Sailboat
A sailmaker is designing a mainsail for a newly constructed sailboat. The mainsail foot is 13 feet long, the luff is 10 feet long, and the luff makes a 52° angle at the head with the leech.
#### Diagram Description
The diagram depicts a triangular mainsail on a sailboat. The three sides of the triangular sail are labeled as follows:
- **Head**: The top vertex of the triangle where the luff and leech meet.
- **Luff**: The side of the triangle closest to the mast (left side), 10 feet long.
- **Leech**: The side opposite to the mast (right side).
- **Foot**: The bottom side of the triangle, 13 feet long.
A 52° angle is shown between the luff and the leech at the head of the sail.
### Calculations
#### Part A: Determine the length of the leech (c).
To determine the length of the leech, we can use the Law of Cosines:
\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]
where
\(a = 10 \) feet (luff),
\(b = 13 \) feet (foot), and
\(C = 52° \).
By substituting these values into the formula:
\[ c^2 = 10^2 + 13^2 - 2(10)(13) \cos(52°) \]
\[ c^2 = 100 + 169 - 2(10)(13) \cos(52°) \]
\[ c^2 = 269 - 260 \cos(52°) \]
Calculate the value of \( \cos(52°) \):
\[ \cos(52°) \approx 0.6157 \]
Substituting this back into the equation:
\[ c^2 = 269 - 260 (0.6157) \]
\[ c^2 = 269 - 160.082 \]
\[ c^2 = 108.918 \]
\[ c \approx \sqrt{108.918} \]
\[ c \approx 10.43 \text{ feet} \]
So, the leech is approximately 10.43 feet long.
#### Part B: Find the area of the mains
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