/*! 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 43 The side lengths of any triangle... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 43

The side lengths of any triangle are related by the Law of cosines, $$c^{2}=a^{2}+b^{2}-2 a b \cos \theta.$$ a. Estimate the change in the side length \(c\) when \(a\) changes from \(a=2\) to \(a=2.03, b\) changes from \(b=4.00\) to \(b=3.96\) and \(\theta\) changes from \(\theta=\pi / 3\) to \(\theta=\pi / 3+\pi / 90.\) b. If \(a\) changes from \(a=2\) to \(a=2.03\) and \(b\) changes from \(b=4.00\) to \(b=3.96,\) is the resulting change in \(c\) greater in magnitude when \(\theta=\pi / 20\) (small angle) or when \(\theta=9 \pi / 20\) (close to a right angle)?

Short Answer

Expert verified
Question: Estimate the change in side length \(c\) of a triangle when side lengths \(a\) and \(b\) change from \(2\) to \(2.03\) and from \(4\) to \(3.96\), respectively, and when the angle \(\theta\) between the two sides changes from \(\frac{\pi}{3}\) to \(\frac{\pi}{3} + \frac{\pi}{90}\). Also, determine if the change in \(c\) would be greater when \(\theta\) is replaced by \(\frac{\pi}{20}\) or \(\frac{9\pi}{20}\). Answer: To estimate the change in side length \(c\), follow the steps in the provided solution. Once you have calculated the change in \(c\) for the given values and for the two cases in part (b), compare the magnitudes of the changes to determine which scenario results in a greater change.

Step by step solution

01

(Step 1: Calculate the initial value of \(c\))

Using the Law of Cosines, we will find the initial value of \(c\) when \(a=2\), \(b=4\), and \(\theta=\frac{\pi}{3}\). The formula is: \(c^2=a^2+b^2-2ab\cos(\theta)\) Substitute the given initial values, and we get: \(c^2=2^2+4^2-2(2)(4)\cos\left(\frac{\pi}{3}\right)\) Calculate and find the value of \(c\).
02

(Step 2: Calculate the new value of \(c\))

Now, we need to find the value of \(c\) after the changes (\(a=2.03, b=3.96\) and \(\theta = \frac{\pi}{3} + \frac{\pi}{90}\)): \(c^2=2.03^2+3.96^2-2(2.03)(3.96) \cos\left(\frac{\pi}{3}+\frac{\pi}{90}\right)\) Calculate and find the new value of \(c\).
03

(Step 3: Estimate the change in \(c\))

Subtract the initial value of \(c\) from the new value of \(c\) to estimate the change in side length \(c\).
04

(Step 4: Calculate the changes in \(c\) for different angles in part b)

To compare the change in \(c\) when \(\theta=\frac{\pi}{20}\) and \(\theta=\frac{9\pi}{20}\), we will calculate the value of \(c\) by substituting these values along with the changed values of \(a\) and \(b\), then subtract the initial value of \(c\) to find the change in each case. Case 1: \(\theta=\frac{\pi}{20}\) \(c^2=2.03^2+3.96^2-2(2.03)(3.96)\cos\left(\frac{\pi}{20}\right)\) Calculate the change in \(c\). Case 2: \(\theta=\frac{9\pi}{20}\) \(c^2=2.03^2+3.96^2-2(2.03)(3.96)\cos\left(\frac{9\pi}{20}\right)\) Calculate the change in \(c\).
05

(Step 5: Compare the changes in \(c\) for both cases in part b)

Now, compare the magnitude of changes in \(c\) for both cases in part (b) to determine if the resulting change in \(c\) is greater when \(\theta=\frac{\pi}{20}\) or when \(\theta=\frac{9\pi}{20}\).

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!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Triangle geometry
The Law of Cosines is a crucial part of triangle geometry, especially when dealing with finding unknown side lengths or angles in non-right triangles. It bridges the gap in scenarios where traditional right-angle trigonometry doesn鈥檛 apply. In any triangle with sides labeled as \(a\), \(b\), and \(c\), and an angle \(\theta\) opposite side \(c\), the Law of Cosines gives the relationship:
  • \(c^2 = a^2 + b^2 - 2ab\cos(\theta)\)
This equation generalizes the Pythagorean theorem by including the angle \(\theta\), which makes it applicable to all types of triangles.
In the exercise, the goal is to evaluate how small changes in the triangle's side lengths \(a\) and \(b\), as well as angle \(\theta\), affect the third side \(c\). This process requires not just understanding the geometry itself, but also how sensitive the side length \(c\) is to changes in the input parameters of the triangle.
Trigonometric functions
Trigonometric functions, particularly the cosine function, play a vital role in the Law of Cosines. The function \(\cos(\theta)\), where \(\theta\) is the angle opposite to side \(c\), determines a part of the value for \(c^2\) and, hence, \(c\) itself.
  • The cosine of an angle relates to the adjacent side and hypotenuse in a right triangle, but here, in any triangle, it provides crucial information about the side's contributions in a non-right scenario.
  • The cosine function varies between -1 and 1, depending on the angle's measurement, and significantly influences the length of \(c\) due to its multiplicative interaction with \(a\) and \(b\).
Changes in \(\theta\) can cause noticeable differences in the calculated length of \(c\), as seen when small angle adjustments are made in the exercise.
Understanding how trigonometric functions like cosine manage angle effects helps in predicting and calculating side changes accurately.
Angle adjustments
Making angle adjustments contributes to the complexity and variability of a triangle鈥檚 geometry. Small variations in the angle \(\theta\) can lead to substantial changes in the side length \(c\).
In the given exercise, this is illustrated by changing \(\theta\) from \(\frac{\pi}{3}\) to \(\frac{\pi}{3} + \frac{\pi}{90}\). The new angle is slightly bigger, and this impacts the length of side \(c\) once recalculated using the Law of Cosines.
  • For smaller angles, closer to zero, these adjustments result in a minor shift because the cosine of small angles is close to 1, reducing the subtracted term's effect in the Law of Cosines formula.
  • Larger angles, however, can cause the cosine value to decrease, thereby increasing the impact and overall length of the side \(c\).
Adjusting angles helps in understanding the relationship between geometric transformations and their algebraic outcomes on triangle dimensions.
This concept is pivotal in solving parts of the exercise and determining the most significant angle scenarios for side length changes.

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

Consider the curve \(\mathbf{r}(t)=\langle\cos t, \sin t, c \sin t\rangle\) for \(0 \leq t \leq 2 \pi,\) where \(c\) is a real number. a. What is the equation of the plane \(P\) in which the curve lies? b. What is the angle between \(P\) and the \(x y\) -plane? c. Prove that the curve is an ellipse in \(P\).

The density of a thin circular plate of radius 2 is given by \(\rho(x, y)=4+x y .\) The edge of the plate is described by the parametric equations \(x=2 \cos t, y=2 \sin t,\) for \(0 \leq t \leq 2 \pi\) a. Find the rate of change of the density with respect to \(t\) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum?

Potential functions arise frequently in physics and engineering. A potential function has the property that \(a\) field of interest (for example, an electric field, a gravitational field, or a velocity field is the gradient of the potential (or sometimes the negative of the gradient of the potential). (Potential functions are considered in depth in Chapter 14 .) In two dimensions, the motion of an ideal fluid (an incompressible and irrotational fluid) is governed by a velocity potential \(\varphi .\) The velocity components of the fluid, \(u\) in the \(x\) -direction and \(v\) in the \(y\) -direction, are given by \(\langle u, v\rangle=\nabla \varphi .\) Find the velocity components associated with the velocity potential \(\varphi(x, y)=\sin \pi x \sin 2 \pi y\).

A clothing company makes a profit of \(\$ 10\) on its long-sleeved T-shirts and \(\$ 5\) on its short-sleeved T-shirts. Assuming there is a \(\$ 200\) setup cost, the profit on \(\mathrm{T}\) -shirt sales is \(z=10 x+5 y-200,\) where \(x\) is the number of long-sleeved T-shirts sold and \(y\) is the number of short-sleeved T-shirts sold. Assume \(x\) and \(y\) are nonnegative. a. Graph the plane that gives the profit using the window $$ [0,40] \times[0,40] \times[-400,400] $$ b. If \(x=20\) and \(y=10,\) is the profit positive or negative? c. Describe the values of \(x\) and \(y\) for which the company breaks even (for which the profit is zero). Mark this set on your graph.

Find the maximum value of \(x_{1}+x_{2}+x_{3}+x_{4}\) subject to the condition that \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=16\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation

Applied Mathematics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

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.