Two sides of a triangle are 14 and 9. Find the size of the angle 0 (in radians) formed by the sides that will maximize the area of the triangle.
Two sides of a triangle are 14 and 9. Find the size of the angle 0 (in radians) formed by the sides that will maximize the area of the triangle.
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Ratios
A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.
Trigonometric Ratios
Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.
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![**Problem Statement: Maximizing the Area of a Triangle**
Two sides of a triangle are 14 and 9. Find the size of the angle \( \theta \) (in radians) formed by the sides that will maximize the area of the triangle.
**Explanation and Approach:**
To solve this problem, we can use the formula for the area of a triangle when two sides and the included angle are given:
\[ \text{Area} = \frac{1}{2}ab \sin(\theta), \]
where \( a \) and \( b \) are the lengths of the two sides, and \( \theta \) is the angle between them.
To maximize the area, we need to maximize \( \sin(\theta) \), since \( \frac{1}{2}ab \) is constant given fixed side lengths. The function \( \sin(\theta) \) reaches its maximum value of 1 when \( \theta = \frac{\pi}{2} \) radians.
Therefore, to maximize the area of the triangle with side lengths 14 and 9, the angle \( \theta \) should be \( \frac{\pi}{2} \) radians, which corresponds to a right angle.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8084badb-1caf-4ac3-abaa-e998705ca4db%2F2feb9b38-f871-4582-b395-88d6abaeb1ef%2Fem0a0fm_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement: Maximizing the Area of a Triangle**
Two sides of a triangle are 14 and 9. Find the size of the angle \( \theta \) (in radians) formed by the sides that will maximize the area of the triangle.
**Explanation and Approach:**
To solve this problem, we can use the formula for the area of a triangle when two sides and the included angle are given:
\[ \text{Area} = \frac{1}{2}ab \sin(\theta), \]
where \( a \) and \( b \) are the lengths of the two sides, and \( \theta \) is the angle between them.
To maximize the area, we need to maximize \( \sin(\theta) \), since \( \frac{1}{2}ab \) is constant given fixed side lengths. The function \( \sin(\theta) \) reaches its maximum value of 1 when \( \theta = \frac{\pi}{2} \) radians.
Therefore, to maximize the area of the triangle with side lengths 14 and 9, the angle \( \theta \) should be \( \frac{\pi}{2} \) radians, which corresponds to a right angle.
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