009A Sample Final 2, Problem 5 - Revision history

MathAdmin at 19:36, 1 December 2017

2017-12-01T19:36:23Z

MathAdmin at 19:35, 1 December 2017

2017-12-01T19:35:54Z

MathAdmin at 19:50, 26 May 2017

2017-05-26T19:50:19Z

MathAdmin at 19:48, 26 May 2017

2017-05-26T19:48:28Z

MathAdmin at 19:12, 25 May 2017

2017-05-25T19:12:10Z

MathAdmin: Created page with " A lighthouse is located on a small island 3km away from the nearest point $P$ on a straight shoreline an..."

2017-04-10T16:23:33Z

Created page with "<span class="exam"> A lighthouse is located on a small island 3km away from the nearest point <math style="vertical-align: 0px">P</math> on a straight shoreline an..."

New page

<span class="exam"> A lighthouse is located on a small island 3km away from the nearest point  <math style="vertical-align: 0px">P</math>  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  <math style="vertical-align: 0px">P?</math>

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Foundations:  
|-
|
|-
|
|-
|
|-
|
|}

'''Solution:'''

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 1:  
|-
|
|-
|
|-
|
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Step 2:  
|-
|
|-
|
|}

{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
!Final Answer:  
|-
|
|}
[[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 88: / Line 88: @@
 & = & \displaystyle{3\left(\frac{\sqrt{10}}{3}\right)^{2}(8\pi)}\\
 &&\\
-& = & \displaystyle{3\left(\frac{10}{3}\right)(8\pi)}\\
+& = & \displaystyle{3\left(\frac{10}{9}\right)(8\pi)}\\
 &&\\
-& = & \displaystyle{80\pi\text{ km/min.}}
+& = & \displaystyle{\frac{80\pi}{3} \text{ km/min.}}
 \end{array}</math>
 |}

@@ Line 23: / Line 23: @@
 |We can begin this physical word problem by drawing a picture.
 |-
-|[[File:009A_SF2_5_GP.png|center|325px]]
+|[[File:009A_SF2_5GP.png|left|400px]]
 |-
 |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>

← Older revision		Revision as of 19:12, 25 May 2017
Line 1:		Line 1:
−	<span class="exam"> A lighthouse is located on a small island ~~3km~~ away from the nearest point  <math style="vertical-align: 0px">P</math>  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  <math style="vertical-align: 0px">P?</math>	+	<span class="exam"> A lighthouse is located on a small island 3 km away from the nearest point  <math style="vertical-align: 0px">P</math>  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  <math style="vertical-align: 0px">P?</math>

	{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"		{\| class="mw-collapsible mw-collapsed" style = "text-align:left;"

@@ Line 98: / Line 98: @@
 !Final Answer: &nbsp;
 |-
-|&nbsp; &nbsp; &nbsp; &nbsp;<math>80\pi\text{ km/min}</math>
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>\frac{80\pi}{3} \text{ km/min}</math>
 |}
 [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]

@@ Line 1: / Line 1: @@
-<span class="exam"> A lighthouse is located on a small island 3 km away from the nearest point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; 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 &nbsp;<math style="vertical-align: 0px">P?</math>
+<span class="exam"> A lighthouse is located on a small island 3km away from the nearest point &nbsp;<math style="vertical-align: 0px">P</math>&nbsp; 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 &nbsp;<math style="vertical-align: 0px">P?</math>
 {| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 !Foundations: &nbsp;
 |-
+|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
 |}
@@ Line 19: / Line 21: @@
 !Step 1: &nbsp;
 |-
+|We can begin this physical word problem by drawing a picture.
 |-
+|By drawing a right triangle with the beam as its hypotenuse, we can see that our variable
 |-
+|&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.
 |}
@@ Line 31: / Line 38: @@
 !Step 2: &nbsp;
 |-
+|Now, we use implicit differentiation to find
 |-
+|
 |}
@@ Line 40: / Line 98: @@
 !Final Answer: &nbsp;
 |-
+|&nbsp; &nbsp; &nbsp; &nbsp;<math>80\pi\text{ km/min}</math>
 |}
 [[009A_Sample_Final_2|'''<u>Return to Sample Exam</u>''']]