L1.4 B1 Solving Exponential Equations - Part 4
Completion requirements
Unit 1
Functions
Sometimes graphing technology is needed.
The compound interest formula is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mo»=«/mo»«mi»P«/mi»«msup»«mfenced»«mrow»«mi»l«/mi»«mo»+«/mo»«mi»i«/mi»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mrow»«/mstyle»«/math».
Determine how many years an investment of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»$«/mi»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«/mrow»«/mstyle»«/math» must be in an account that earns «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»3«/mn»«mo»%«/mo»«/mrow»«/mstyle»«/math» compounded semi-annually before it increases in value to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»$«/mi»«mn»1«/mn»«mo»§#160;«/mo»«mn»490«/mn»«/mstyle»«/math».
Determine how many years an investment of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi mathvariant=¨normal¨»$«/mi»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«/mrow»«/mstyle»«/math» must be in an account that earns «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»8«/mn»«mo».«/mo»«mn»3«/mn»«mo»%«/mo»«/mrow»«/mstyle»«/math» compounded semi-annually before it increases in value to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi mathvariant=¨normal¨»$«/mi»«mn»1«/mn»«mo»§#160;«/mo»«mn»490«/mn»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»P«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»principle«/mi»«mspace width=¨0.33em¨/»«mi»amount«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»I«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»interest«/mi»«mspace width=¨0.33em¨/»«mi»rate«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»8«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»3«/mn»«mi mathvariant=¨normal¨»%«/mi»«mo»§#247;«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mo»(«/mo»«mi»because«/mi»«mo»§#160;«/mo»«mi»it«/mi»«mo»§#160;«/mo»«mi»is«/mi»«mo»§#160;«/mo»«mi»compounded«/mi»«mo»§#160;«/mo»«mi»semi«/mi»«mo»-«/mo»«mi»annually«/mi»«mo»,«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mi»twice«/mi»«mo»§#160;«/mo»«mi»per«/mi»«mo»§#160;«/mo»«mi»year«/mi»«mo»)«/mo»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»4«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»15«/mn»«mi mathvariant=¨normal¨»%«/mi»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»0415«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»A«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»490«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Now, replace all the known values in the formula.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨left¨»«mtr»«mtd»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»490«/mn»«mo»=«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mo»+«/mo»«mn»0«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»0415«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mtd»«/mtr»«mtr»«mtd»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»490«/mn»«mo»=«/mo»«mn»1«/mn»«mspace width=¨0.33em¨/»«mn»000«/mn»«msup»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»0415«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Now, divide both sides of the equation by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mo»§#160;«/mo»«mn»000«/mn»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»490«/mn»«mo»=«/mo»«msup»«mfenced»«mrow»«mn»1«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»0415«/mn»«/mrow»«/mfenced»«mi»n«/mi»«/msup»«/mrow»«/mstyle»«/math»
Graph both sides of the equation as separate functions, and then find the point of intersection of the two graphs.
The graphs intersect at approximately «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»n«/mi»«mo»=«/mo»«mn»9«/mn»«mi mathvariant=¨normal¨».«/mi»«mn»8«/mn»«/mrow»«/mstyle»«/math».
Compounding periods are whole numbers. As such, it will take «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» compounding periods before the value of the investment reaches «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mo»$«/mo»«mn»1«/mn»«mo»§#160;«/mo»«mn»490«/mn»«/mstyle»«/math».
Because interest was compounded semi-annually, the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math» compounding periods are equivalent to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»5«/mn»«/mstyle»«/math» years.