Difference between revisions of "009A Sample Final 2, Problem 5"
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|We want to know the rate that the beam is moving along the shore when | |We want to know the rate that the beam is moving along the shore when | ||
| − | we are one km away from the point <math>P.</math> | + | we are one km away from the point <math style="vertical-align: 0px">P.</math> |
|- | |- | ||
| − | |This tells us that <math>x=1.</math> | + | |This tells us that <math style="vertical-align: -1px">x=1.</math> |
|- | |- | ||
|The problem also tells us that the lighthouse beam is revolving at 4 revolutions | |The problem also tells us that the lighthouse beam is revolving at 4 revolutions | ||
per minute. | per minute. | ||
|- | |- | ||
| − | |However, <math>\theta</math> is measured in radians, and there are | + | |However, <math style="vertical-align: 0px">\theta</math> is measured in radians, and there are <math style="vertical-align: 0px">2\pi</math> radians in a revolution. |
| − | <math>2\pi</math> radians in a revolution. | ||
|- | |- | ||
|Thus, we know | |Thus, we know | ||
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| <math>{\displaystyle \frac{d\theta}{dt}\ =\ 4\cdot2\pi\ =\ 8\pi.}</math> | | <math>{\displaystyle \frac{d\theta}{dt}\ =\ 4\cdot2\pi\ =\ 8\pi.}</math> | ||
|- | |- | ||
| − | |Finally, we require secant. Since we know <math>x=1,</math> | + | |Finally, we require secant. Since we know <math style="vertical-align: -3px">x=1,</math> |
|- | |- | ||
|we can solve the triangle to get that the length of the hypotenuse is | |we can solve the triangle to get that the length of the hypotenuse is | ||
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|- | |- | ||
|This means that | |This means that | ||
| − | | | + | |- |
| − | | | + | | <math>{\displaystyle \sec\theta\ =\ \frac{1}{\cos\theta}\ =\ \frac{\textrm{hyp.}}{\textrm{side adj.}}\ =\ \frac{\sqrt{10}}{3}.}</math> |
|} | |} | ||
Revision as of 10:42, 26 May 2017
A lighthouse is located on a small island 3km away from the nearest 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} 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 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?}
| 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: |
|---|
| Now, we use implicit differentiation to find |
| 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 \sec^{2}\theta\cdot\frac{d\theta}{dt}\ =\ \frac{1}{3}\cdot\frac{dx}{dt}.}} |
| Rearranging, we have |
| 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 \frac{dx}{dt}\ =\ 3\sec^{2}\theta\cdot\frac{d\theta}{dt}.}} |
| Again, everything is a function of time. |
| Step 3: |
|---|
| We want to know the rate that the beam is moving along the shore when
we are one km away 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.} |
| This tells us that 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=1.} |
| The problem also tells us that the lighthouse beam is revolving at 4 revolutions
per minute. |
| However, 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} is measured in radians, and there are 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 2\pi} radians in a revolution. |
| Thus, we know |
| 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 \frac{d\theta}{dt}\ =\ 4\cdot2\pi\ =\ 8\pi.}} |
| Finally, we require secant. Since we know 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=1,} |
| we can solve the triangle to get that the length of the hypotenuse is |
| 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 \sqrt{1^{2}+3^{2}}=\sqrt{10}.} |
| This means that |
| 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 \sec\theta\ =\ \frac{1}{\cos\theta}\ =\ \frac{\textrm{hyp.}}{\textrm{side adj.}}\ =\ \frac{\sqrt{10}}{3}.}} |
| Step 4: |
|---|
| Now, we can plug in all these values to find |
|
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 \begin{array}{rcl} \displaystyle{\frac{dx}{dt}} & = & \displaystyle{3\sec^{2}\theta\cdot\frac{d\theta}{dt}}\\ &&\\ & = & \displaystyle{3\left(\frac{\sqrt{10}}{3}\right)^{2}(8\pi)}\\ &&\\ & = & \displaystyle{3\left(\frac{10}{3}\right)(8\pi)}\\ &&\\ & = & \displaystyle{80\pi\text{ km/min.}} \end{array}} |
| Final Answer: |
|---|
| 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 80\pi\text{ km/min}} |