/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 25 What is the largest possible are... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 5: Problem 25

What is the largest possible area for a triangle that has one side of length 4 and one side of length \(7 ?\)

Short Answer

Expert verified
The largest possible area for a triangle with sides of length 4 and 7 is 14 square units, which occurs when the angle between these sides is \(90^\circ\), making the triangle a right-angled triangle.

Step by step solution

01

Identify the variables

The given sides of the triangle are length 4 and 7, say \(a=4\) and \(b=7\), respectively. Let \(\theta\) be the angle between these sides and c be the third side. To find the maximum area, we need to find the optimal angle \(\theta\).
02

Area of the triangle

To find the area A of the triangle, we use the formula involving the sine of the angle between the given sides: \[A = \frac{1}{2}ab\sin\theta\] In our case, this becomes: \[A = \frac{1}{2}(4)(7)\sin\theta\] So, \(A = 14\sin\theta\).
03

Using sine rule to find angle

We will now apply the sine rule to find a relationship between the angle \(\theta\) and its opposite side c. The sine rule states: \[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\] In our case, we know that \(c = 4\), so we have: \[\frac{7}{\sin\theta} = \frac{4}{\sin(180^\circ-\theta)}\]
04

Find the optimal angle

Our goal is to find the value of \(\theta\) such that the area A is maximized. By using properties of sine function, we know that: \[\sin\theta = \sin(180^\circ-\theta)\] The function sine takes its maximum value when its argument is 90 degrees. Thus, to maximize area A, we should choose \(\theta = 90^\circ\), which would make the triangle a right-angled triangle.
05

Calculate the maximum area

Using the optimal angle \(\theta = 90^\circ\) in the triangle area formula, we get: \[A = 14 \sin(90^\circ) = 14\] Thus, the largest possible area for the triangle is 14 square units.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the smallest positive number \(\theta\) such that \(10^{\cos \theta}=6 .\)

Show that $$|\cos x+\sin x| \leq \sqrt{2}$$ for every number \(x\).

Show that $$\cos \frac{\pi}{32}=\frac{\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}}{2}$$ [Hint: First do Exercise 66.]

Evaluate the given quantities assuming that \(u\) and \(v\) are both in the interval \(\left(-\frac{\pi}{2}, 0\right)\) and \(\tan u=-\frac{1}{7} \quad\) and \(\quad \tan v=-\frac{1}{8}\) $$\sin (2 v)$$

Explain how the equation \(\sin \frac{3 \pi}{10}=\frac{\sqrt{5}+1}{4}\) follows from the solution to Exercise \(9 .\)
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Statistics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.