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7 Exponential Growth
7 Exponential Growth
Y = Answer, a = Starting Value, b = Muliplier, x = Time
Predicting the Future
Predicting the Future
One can use the exponential formula to predict future values.
For example, let's say a gallon of milk costs $4.50 today. We will presume that inflation is 3%. Therefore, our multiplier = 1.03.
What will a gallon of milk cost 6 years in the future?
Start with the formula: y = ab^x
Plug in the values: y = $4.50 * 1.03^6
Use a computer or Desmos to calculate.
y = $5.37 will be the cost of a gallon of milk six years in the future!
Calculating the Past
Calculating the Past
One can use the exponential formula to calculate past values.
For example, let's say a gallon of milk costs $4.50 today. We will presume that inflation is 3%. Therefore, our multiplier = 1.03.
What did a gallon of mike cost 10 years ago?
Start with the formula: y = ab^x
Plug in values: y = $4.50 * 1.03^(-10)
Use a computer or Desmos to calculate.
y = $3.35 is the cost of a gallon of milk ten years ago!
Interest rates and multipliers
Interest rates and multipliers
If the interest rate is 1%, the multiplier equals 1.01.
If the interest rate is 1.5%, the multiplier equals 1.015.
If the interest rate is 10%, the multiplier equals 1.10.
If the interest rate is 17.6%, the multiplier equals 1.176.
Depreciation
Depreciation
If something is losing 5% of its value, the multiplier equals 0.95.
If something is losing 12% of its value, the multiplier equals 0.88.
Calculating The Multipliers!
Calculating The Multipliers!
Consider the table to the right. What is the multiplier? What is b?
We start with 100. We multiply 100 by b and get 80. Solve for b!
100x = 80
x = 80/100 = 0.80. Our multiplier, b equals 0.80.
Calculating the Starting Number!
Calculating the Starting Number!
Calculating For a!
Calculating For a!
What is the value of a at day 0 in the table to the right?
What is the value of a at day 0 in the table to the right?
From the previous problem ... we know that the b = 0.8 we can calculate a.
From the previous problem ... we know that the b = 0.8 we can calculate a.
a * 0.8 = 100 To get a by itself, divide both sides of the equation by 0.8
a = 100/0.8
a = 125! Our starting number is 125!
How to calculate an exponential equation from data.
How to calculate an exponential equation from data.
Let's say that on day 2 there were 9 zombies.
By day 5 there were 243 zombies.
What is the exponential equation?
To solve this problem I will write the y = ab^x formula. Then I will write the two equations under it.
y = ab^x Now I will plug in the y values. I will put the larger y value on top of the other equation.
243 = ab^5 I wrote the bigger numbers on top.
9 = ab^2 Divide everything. The a variable disappears as a 1.
27 = b^3
Now I need to figure out the cube root of 27. I could ask Google or I could type 27^(1/3) in the computer.
b = 3! The multiplier is 3!
Now that I know b, I can use it to calculate a! I will start by writing the formula again:
y = ab^x
9 = a * 3^2 I plug in values that I know. I choose to use the easier equation.
9 = a * 9 Obviously, 9 = 1 * 9.
a = 1.
The entire equation: y = 1 * 3^x
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