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Copyright 2014 Scott Storla Formulas
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Copyright 2014 Scott Storla
Formulas
Copyright 2014 Scott Storla
A formula is an equation that uses mathematics to express a fact or rule.
Usually the left hand side is a single letter that represents the concept we’re working with. The right hand side might include numbers, variables, operators and grouping symbols.
Copyright 2014 Scott Storla
The Pythagorean Theorem
The Pythagorean Theorem
English – Given a right triangle the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse.
Algebra – If a and b are the lengths of the legs of a right triangle and cis the length of the hypotenuse then 2 2 2a b c .
a
b
c
2 2 2a b c
Copyright 2014 Scott Storla
2 2 2a b c
Find the length of the missing leg.
b feet
2 225 13b
225 169b
Oper Inv
^2
25 25
2 144b
144 12b
The length of the missing leg is 12 feet.
Copyright 2014 Scott Storla
One leg of a right triangle is 4 inches shorter than the leg that is 20 inches. Find the length of the hypotenuse.
2 2 2a b c
2 2 216 20 c
2256 400 c
2656 c
656 c
4 41 c
25.6 c
656
2 328
2 2 164
2 2 2 82
2 2 2 2 41
The length of the hypotenuse is about 25.6 inches.
Copyright 2014 Scott Storla
2 2 2a b c
2 2 216 12 c
2256 144 c
2400 c
400 c
20 c
The Pythagorean Theorem can help us decide if something we build is “square” (the corners are 90 degrees). You’ve built a deck that’s 12 feet by 16 feet. If the deck is square what should the diagonal distance from corner to corner through the center of the deck be?
16 feet
12 feet
c feet
The length of the diagonal should be 20 feet.
Copyright 2014 Scott Storla
Temperature
932
5F C
Would a hot tub that’s 60º Celsius be too hot or too cold?
932
5F C
960 32
5F
108 32F
140F
It would be too hot.
Copyright 2014 Scott Storla
Temperature
932
5F C
Water freezes at 32º Fahrenheit. At what Celsius temperature does water freeze.
932
5F C
932 32
5C
90
5C
0 C
Water freezes at 0° Celsius.
Oper Inv
9 5
5 932 32
5 5 90
9 9 5C
Copyright 2014 Scott Storla
Temperature
932
5F C
A passenger from Canada says they prefer the temperature in your car to be around 20 degrees. To accommodate their needs what Fahrenheit temperature should you use?
920 32
5F
9 4 32F
36 32F
You should set the temperature at 68° Fahrenheit.
68F
932
5F C
Copyright 2014 Scott Storla
Temperature
932
5F C
On a trip to Europe you plan to cook a dish for your relatives and you need to set the oven to 350º Fahrenheit. What Celsius temperature should you use?
932
5F C
9350 32
5C
9318
5C
177 C
You should set the temperature of the oven to 177° C.
Oper Inv
9 5
5 932 32
5 5 9318
9 9 5C
0.8 0.8e e
Copyright 2014 Scott Storla
Continuous Compounding
rtA Pe
How much should be saved today to have $50,000 in 20 years at 4% interest compounded continuously?
0.04 2050,000 Pe
0.850,000 Pe
22,467 P
I would need to save at least $22,467 today.
rtA Pe
0.01 0.01
10,000 10,000
Copyright 2014 Scott Storla
Continuous Compounding
rtA Pe
How long will it take to double $10,000 at 1% interest?
0.0120,000 10,000 te
69.3 t
It will take about 63 years to double my money at 1%.
rtA Pe
Oper Inv
0.01 0.01
^ ln
10,000 10,000
e
0.012 te0.01ln(2) ln( )te
ln(2) 0.01t
8 8
750,000 750,000
Copyright 2014 Scott Storla
Continuous Compounding
rtA Pe
Currently an account has $750,000. Once the account reaches $1,250,000 the account holder has decided to retire. If they would like to retire in 8 years what interest rate would they need?
81,250,000 750,000 re
0.0639 t
The rate needs to be at least 6.4%.
rtA Pe
Oper Inv
8 8
^ ln
750,000 750,000
e
85
3re
85ln( ) ln( )3re
5ln( ) 83 r
Copyright 2014 Scott Storla
The formula 2 13H t predicts the percent of one-person households in the United States since 1960.
b. Replace H with 20 to predict when the percent of one-person households first rose to 20%.
a. Replace t with 40 to approximate the percent of one-person households in 2000.
c. Replace H with 30 to predict the first year the number of one-person households will reach 30%.
a. Around 25.6% of households will be one person households in 2000.
b. In the year 1972.
c. In 2032 the number of one-person households will first reach 30%.
Copyright 2014 Scott Storla
An organization makes money by selling cookies door to door.
The formula 2
0.75 4.2 10.83P p
describes the profit (P)
made per house if they set the price of a box of their cookies to p dollars.
Find their profit per house if they charge $5.00 per box.
20.75 4.2 10.83P p
20.75 5 4.2 10.83P
10.35P
If they set the price at $5 per box they can expect to make about $10.35 per house.
Copyright 2014 Scott Storla
An organization makes money by selling cookies door to door.
The formula 2
0.75 4.2 10.83P p
describes the profit (P)
made per house if they set the price of a box of their cookies to p dollars.
Find their profit per house if they lower their price to $3.40 per box.
20.75 4.2 10.83P p
20.75 3.4 4.2 10.83P
10.35P
If they set the price at $3.40 per box they can again expect to make about $10.35 per house.
0.75 0.75
Copyright 2014 Scott Storla
An organization makes money by selling cookies door to door.
The formula 2
0.75 4.2 10.83P p
describes the profit (P)
made per house if they set the price of a box of their cookies to p dollars.
The maximum profit the organization can expect per house is $10.83. Set P to 10.83 and solve for p to find the price per box that maximizes profit?
20.75 4.2 10.83P p
210.83 0.75 4.2 10.83p
They will earn the maximum profit per house if they set their price per box at $4.20.
Oper Inv
4.2 4.2
^2
0.75 0.75
10.83 10.83
10.83 10.83
20 0.75 4.2p
20 4.2p
0 4.2p
0 4.2p
4.2 4.2
4.2 p
Copyright 2014 Scott Storla
Formulas
