/*! 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 A high fountain of water is loca... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 43

A high fountain of water is located at the center of a circular pool as shown in Figure \(\mathrm{P} 1.43 .\) Not wishing to get his feet wet, a student walks around the pool and measures its circumference to be \(15.0 \mathrm{~m} .\) Next, the student stands at the edge of the pool and uses a protractor to gauge the angle of elevation at the bottom of the fountain to be \(55.0^{\circ}\). How high is the fountain?

Short Answer

Expert verified
The height of the fountain is approximately \(h=\frac{15.0}{2\pi} \cdot tan(55.0)\) meters.

Step by step solution

01

Calculating the Radius

Since the pool is circular, its circumference can be calculated using the formula \(C=2\pi r\), where C is the circumference and r is the radius. We can rearrange this to find r: \(r=\frac{C}{2\pi}\). Substituting the given value of C=15.0m, we find that the radius is \(r=\frac{15.0}{2\pi}\) meters.
02

Calculating the Height

Given that the angle of elevation (theta) is 55.0 degrees, we can use the definition of tangent, which is the ratio of the opposite side to the adjacent side in a right angled triangle. In this case, the height of the fountain is the opposite side (h) and the radius of the pool is our adjacent side (r). So, the tangent of the angle is \(tan(\theta)=\frac{h}{r}\). We can rearrange to solve for h: \(h= r \cdot tan(\theta)\). We substitute the values for r and theta into the equation to get \(h=\frac{15.0}{2\pi} \cdot tan(55.0)\).
03

Approximation

The equation from the previous step could give an exact value for the height of the fountain if we plug an exact value of \(\pi\) and calculate \(tan(55.0)\). However, generally, we convert the angle to radians and use an approximate value of \(\pi=3.142\) to facilitate the calculation. We then use a calculator to find \(tan(55.0)\) and simplify the expression to give the height in meters.

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.

Circumference calculation
To understand how to calculate the circumference, imagine walking around the border of a circular pool. You can determine the distance you walked by measuring the circumference of the pool. The formula for the circumference of a circle is given by \( C = 2 \pi r \), where \( C \) is the circumference and \( r \) is the radius of the circle.
This formula shows that the circumference is directly proportional to the radius. Therefore, when you know one, you can find the other by rearranging the formula. In the given problem, the measured circumference is \( 15.0 \) meters. By rearranging the formula to solve for the radius, \( r = \frac{C}{2\pi} \), you can substitute \( C = 15.0 \) meters to find the radius.
This step is crucial as knowing the radius helps with further calculations like determining heights or angles related to circular objects.
Angle of elevation
An angle of elevation is an important concept in trigonometry that helps us understand various perspectives. It's the angle formed when you look upward from the horizontal line to an object above the horizontal level.
In our scenario, as the student stands at the edge of the pool and looks toward the top of the fountain, the student's line of sight forms an angle of elevation with the ground. The protractor shows an angle of elevation of \( 55.0^{\circ} \).
This angle is essential for calculating heights, as it connects the observer's position to the height of the object using trigonometric functions.
  • Given an object and its angle of elevation, these angles are often used to solve problems about heights and distances.
Understanding this angle provides a way to explore our 3D space with more accuracy using mathematics.
Tangent function
When you need to solve problems involving angles and lengths, the tangent function is often your go-to tool. In a right-angled triangle, the tangent of an angle \( \theta \) is defined as the ratio of the opposite side to the adjacent side.
For this exercise, the problem involves a right triangle formed by the radius of the pool, the height of the fountain, and the direct line between them. When we want to find the height of the fountain, we use the tangent function: \( \tan(\theta) = \frac{h}{r} \), where \( h \) is the height, \( r \) is the radius, and \( \theta \) is \( 55.0^{\circ} \).
By rearranging the equation to find \( h \), we get \( h = r \cdot \tan(\theta) \). This step requires using a calculator to find \( \tan(55.0) \), and knowing the radius \( r \), you can compute \( h \).
  • Tangent helps relate angles within triangles to ratios of sides, making it a powerful tool in trigonometry.
With the tangent function, you can swiftly find unknown sides of triangles as long as you have the necessary angles and side lengths.

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

1\. Suppose your hair grows at the rate of \(1 / 32\) inch per day. Find the rate at which it grows in nanometers per second. Because the distance between atoms in a molecule is on the order of \(0.1 \mathrm{~nm}\), your answer suggests how rapidly atoms are assembled in this protein synthesis.

A rectangular plate has a length of \((21.3 \pm 0.2) \mathrm{cm}\) and a width of \((9.8 \pm 0.1) \mathrm{cm}\). Calculate the area of the plate, including its uncertainty.

A woman measures the angle of elevation of a mountaintop as \(12.0^{\circ}\). After walking \(1.00 \mathrm{~km}\) closer to the mountain on level ground, she finds the angle to be \(14.0^{\circ} .\) (a) Draw a picture of the problem, neglecting the height of the woman's eyes above the ground. Hint: Use two triangles. (b) Select variable names for the mountain height (suggestion: \(y\) ) and the woman's original distance from the mountain (suggestion: \(x\) ) and label the picture. (c) Using the labeled picture and the tangent function, write two trigonometric equations relating the two sclected variables. (d) Find the height \(y\) of the mountain by first solving one equation for \(x\) and substituting the result into the other equation.

(a) Find a conversion factor to convert from miles per hour to kilometers per hour. (b) For a while, federal law mandated that the maximum highway speed would be 55 \(\mathrm{mi} / \mathrm{h}\). Use the conversion factor from part (a) to find the speed in kilometers per hour, (c) The maximum highway speed has been raised to \(65 \mathrm{mi} / \mathrm{h}\) in some places. In kilometers per hour, how much of an increase is this over the \(55-\mathrm{mi} / \mathrm{h}\) limit?

A point is located in a polar coordinate system by the coordinates \(r=2.5 \mathrm{~m}\) and \(\theta=35^{\circ}\). Find the \(x\) - and \(y\) coordinates of this point, assuming that the two coordinate systems have the same origin.
See all solutions

Recommended explanations on Physics Textbooks

Magnetism and Electromagnetic Induction

Read Explanation

Magnetism

Read Explanation

Solid State Physics

Read Explanation

Linear Momentum

Read Explanation

Fields in Physics

Read Explanation

Engineering Physics

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.