Difference between revisions of "022 Exam 2 Sample A, Problem 5"

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(Created page with "<span class="exam"> '''Set up the equation to solve. You only need to plug in the numbers - not solve for particular values!''' <span class="exam">How much money would I hav...")
 
 
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::<math>A=P\left(1+\frac{r}{n}\right)^{nt},</math>
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::<math>A\,=\,P\left(1+\frac{r}{n}\right)^{nt},</math>
 
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|where <math style="vertical-align: 0%;">A</math> is the value of the account, <math style="vertical-align: 0%;">P</math> is the principal (original amount invested), <math style="vertical-align: 0%;">r</math> is the annual rate and <math style="vertical-align: 0%;">n</math> is the number of compoundings per year.  The value of <math style="vertical-align: 0%;">n</math> is <math style="vertical-align: 0%;">365</math> for compounding daily, <math style="vertical-align: 0%;">52</math> for compounding weekly, and <math style="vertical-align: -5%;">12</math> for compounding monthly.  As a result, the exponent <math style="vertical-align: 0%;">nt</math>, where <math style="vertical-align: 0%;">t</math> is the time in years, is the number of compounding periods where we actually earn interest. Similarly, <math style="vertical-align: -22%;">r/n</math> is the rate per compounding period (the annual rate divided by the number of compoundings per year).
 
|where <math style="vertical-align: 0%;">A</math> is the value of the account, <math style="vertical-align: 0%;">P</math> is the principal (original amount invested), <math style="vertical-align: 0%;">r</math> is the annual rate and <math style="vertical-align: 0%;">n</math> is the number of compoundings per year.  The value of <math style="vertical-align: 0%;">n</math> is <math style="vertical-align: 0%;">365</math> for compounding daily, <math style="vertical-align: 0%;">52</math> for compounding weekly, and <math style="vertical-align: -5%;">12</math> for compounding monthly.  As a result, the exponent <math style="vertical-align: 0%;">nt</math>, where <math style="vertical-align: 0%;">t</math> is the time in years, is the number of compounding periods where we actually earn interest. Similarly, <math style="vertical-align: -22%;">r/n</math> is the rate per compounding period (the annual rate divided by the number of compoundings per year).
 
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|For example, if we compound monthly for <math style="vertical-align: 0%;">7</math> years at a <math style="vertical-align: -10%;">6%</math> rate, we would have <math style="vertical-align: -21%;">nt\,=\,12(7)\,=\,84</math> months at a rate of <math style="vertical-align: -22%;">0.06/12\,=\,0.005</math>, or <math style="vertical-align: -10%;">0.5%</math>), per monthly period. Notice that we <u>'''always'''</u> use the decimal version for interest rates in these equations.
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|For example, if we compound monthly for <math style="vertical-align: 0%;">7</math> years at a <math style="vertical-align: -5%;">6%</math> rate, we would compound <math style="vertical-align: -5%;">nt\,=\,12 \cdot7\,=\,84</math> times, once per month, at a rate of <math style="vertical-align: -22%;">0.06/12\,=\,0.005</math> per monthly period. Notice that we <u>'''always'''</u> use the decimal version for interest rates when using these equations.
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|On the other hand, interest compounded continuously earns rate in just that way - continuously.  I can have any value I want for time <math style="vertical-align: 0%;">t</math>, and the total amount in the account will change with each and every second.  Therefore, we express the account value as
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::<math>A\,=\,Pe^{rt}.</math>
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|Notice that in both formulas, the value at the initial time <math style="vertical-align: 0%;">t\,=\,0</math> is just our initial investment <math style="vertical-align: 0%;">P</math>. The goal in the two parts of this problem is to choose the correct equation for each.
 
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
 
{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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!Step 3: &nbsp;
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!(a): &nbsp;
 
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|We can now use the chain rule to find<br>
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|We are given all the pieces required.  We begin with <math style="vertical-align: -5%;">$3000</math> of principal, and compound monthly, or <math style="vertical-align: -5%;">12</math> times per year. Using the formula in 'Foundations', the equation for the account value is
 
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::<math>\begin{array}{rcl}
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::<math>A\,=\,P\left(1+\frac{r}{n}\right)^{nt}\,=\,3000\left(1+\frac{0.045}{12}\right)^{12\cdot 6}.</math>
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y' & = & \left(g\circ f\right)'(x)\\
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\\
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& = & g'\left(f(x)\right)\cdot f'(x)\\
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{| class="mw-collapsible mw-collapsed" style = "text-align:left;"
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\\& = & \displaystyle{\frac{x}{(x+5)(x-1)}\cdot\frac{x^{2}-9x+5}{x^{2}}}\\
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!(b): &nbsp;
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\\
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& = & \displaystyle{\frac{x^{2}-9x+5}{x^{3}+4x^{2}-5x}.}
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|Again, we need to apply the formula from foundations to find
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\end{array}</math>
 
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Note that many teachers <u>'''do not'''</u> prefer a cleaned up answer, and may request that you <u>'''do not simplify'''</u>.  In this case, we could write the answer as<br>
 
 
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::<math>y'=\displaystyle {\frac{x}{(x+5)(x-1)}\cdot\frac{(2x+5)x-(x^{2}+4x-5)(1)}{x^{2}}.}</math>
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::<math>A\,=\,Pe^{rt}\,=\,3000e^{0.045\cdot 6}.</math>
  
 
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!Final Answer: &nbsp;
 
!Final Answer: &nbsp;
 
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|<math>y'\,=\,\displaystyle{\frac{x^{2}-9x+5}{x^{3}+4x^{2}-5x}.}</math>
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::(a) <math style="vertical-align: -80%;">A\,=\,3000\left(1+\frac{0.045}{12}\right)^{12\cdot 6}.</math>
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::(b) <math style="vertical-align: 0%;">A\,=\,Pe^{rt}\,=\,3000e^{0.045\cdot 6}.</math>
 
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[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]
 
[[022_Exam_2_Sample_A|'''<u>Return to Sample Exam</u>''']]

Latest revision as of 21:31, 14 May 2015

Set up the equation to solve. You only need to plug in the numbers - not solve for particular values!

How much money would I have after 6 years if I invested $3000 in a bank account that paid 4.5% interest,

(a) compounded monthly?
(b) compounded continuously?
Foundations:  
The primary purpose of this problem is to demonstrate that you understand the difference between continuous compounding and compounding on an interval of time. When we compound on an interval, say monthly, the value in the account only changes at the end of each interval. In other words, there is no interest accrued for a week or a day. As a result, we use the formula
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle A\,=\,P\left(1+{\frac {r}{n}}\right)^{nt},}
where is the value of the account, 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} is the principal (original amount invested), 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 r} is the annual rate and 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 n} is the number of compoundings per year. The value of 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 n} 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 365} for compounding daily, 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 52} for compounding weekly, and 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 12} for compounding monthly. As a result, the exponent 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 nt} , where 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} is the time in years, is the number of compounding periods where we actually earn interest. Similarly, 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 r/n} is the rate per compounding period (the annual rate divided by the number of compoundings per year).
For example, if we compound monthly for 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 7} years at a 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 6%} rate, we would compound 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 nt\,=\,12 \cdot7\,=\,84} times, once per month, at a rate of 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 0.06/12\,=\,0.005} per monthly period. Notice that we always use the decimal version for interest rates when using these equations.
On the other hand, interest compounded continuously earns rate in just that way - continuously. I can have any value I want for time 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} , and the total amount in the account will change with each and every second. Therefore, we express the account value as
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 A\,=\,Pe^{rt}.}
Notice that in both formulas, the value at the initial time 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\,=\,0} is just our initial investment 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} . The goal in the two parts of this problem is to choose the correct equation for each.
(a):  
We are given all the pieces required. We begin with 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 $3000} of principal, and compound monthly, 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 12} times per year. Using the formula in 'Foundations', the equation for the account value 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 A\,=\,P\left(1+\frac{r}{n}\right)^{nt}\,=\,3000\left(1+\frac{0.045}{12}\right)^{12\cdot 6}.}
(b):  
Again, we need to apply the formula from foundations 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 A\,=\,Pe^{rt}\,=\,3000e^{0.045\cdot 6}.}
Final Answer:  
(a) 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 A\,=\,3000\left(1+\frac{0.045}{12}\right)^{12\cdot 6}.}
(b) 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 A\,=\,Pe^{rt}\,=\,3000e^{0.045\cdot 6}.}

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