Difference between revisions of "009A Sample Final 2, Problem 5"

From Grad Wiki
Jump to navigation Jump to search
Line 4: Line 4:
 
!Foundations:    
 
!Foundations:    
 
|-
 
|-
−
|
+
|When we see a problem talking about rates, it is usually a '''related rates''' problem.
 
|-
 
|-
−
|
+
|Thus, we treat everything as a function of time, or &nbsp;<math style="vertical-align: -1px">t.</math>
 
|-
 
|-
−
|
+
|We can usually find an equation relating one unknown to another, and then use implicit differentiation.
 
|-
 
|-
−
|
+
|Since the problem usually gives us one rate, and from the given info we can usually find the values of
 +
variables at our particular moment in time, we can solve the equation
 +
for the remaining rate.
 
|}
 
|}
  
Line 19: Line 21:
 
!Step 1: &nbsp;  
 
!Step 1: &nbsp;  
 
|-
 
|-
−
|
+
|We can begin this physical word problem by drawing a picture.
 
|-
 
|-
−
|
+
|Insert picture
 
|-
 
|-
−
|
+
|In the picture, we can consider the distance from the point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; to the spot the light hits the shore to be the variable &nbsp;<math style="vertical-align: 0px">x.</math>
 
|-
 
|-
−
|
+
|By drawing a right triangle with the beam as its hypotenuse, we can see that our variable
 +
&nbsp;<math style="vertical-align: 0px">x</math>&nbsp; is related to the angle &nbsp;<math style="vertical-align: 0px">\theta</math>&nbsp; by the equation
 +
|-
 +
|&nbsp; &nbsp; &nbsp; &nbsp;<math>{\displaystyle \tan\theta\ =\ \frac{\textrm{side opp.}}{\textrm{side adj. }}\ =\ \frac{x}{3}.}</math>
 +
|-
 +
|This gives us a relation between the two variables.
 
|}
 
|}
  

Revision as of 10:26, 26 May 2017

A lighthouse is located on a small island 3km away from the nearest point    on a straight shoreline and its light makes 4 revolutions per minute. How fast is the beam of light moving along the shoreline on a point 1km away from  

Foundations:  
When we see a problem talking about rates, it is usually a related rates problem.
Thus, we treat everything as a function of time, or  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t.}
We can usually find an equation relating one unknown to another, and then use implicit differentiation.
Since the problem usually gives us one rate, and from the given info we can usually find the values of

variables at our particular moment in time, we can solve the equation for the remaining rate.


Solution:

Step 1:  
We can begin this physical word problem by drawing a picture.
Insert picture
In the picture, we can consider the distance from the point  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P}   to the spot the light hits the shore to be the variable  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x.}
By drawing a right triangle with the beam as its hypotenuse, we can see that our variable

 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}   is related to the angle  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta}   by the equation

       Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\displaystyle \tan\theta\ =\ \frac{\textrm{side opp.}}{\textrm{side adj. }}\ =\ \frac{x}{3}.}}
This gives us a relation between the two variables.
Step 2:  


Final Answer:  

Return to Sample Exam

Retrieved from "https://gradwiki.math.ucr.edu/index.php?title=009A_Sample_Final_2,_Problem_5&oldid=3419"