activity 1 a helicopter takes off from the roof of a building that is 200 feet about the ground. the...
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Activity 1A helicopter takes off from the roof of
a building that is 200 feet about the ground. The altitude of the helicopter increases by 150 feet each minute. Express the height of the helicopter as a function of minutes after it starts rising.
Exponential function
A linear relationship is one in which there is a fixed rate of change (slope).
An exponential relationship is one in which for a fixed change in x, there is a fixed percent change in y.
Percent ChangeLet’s say I have 10 apples. If someone gives me 1 apple then I’ve
increased my apples to 11 apples. Old value = 10, New Value = 11 I can also say I’ve increased my apple
supply by what percent?
11 10.1 or 10%
10
Let’s say I have $10 and someone steals $2. By what percent has my dollar amount changed?
Can we come up with a general formula for percent change?
new old new reference
ercent Changeold reference
P
8 10 2.2 or 20%
10 10
What if I said I had 10 apples and I increased my number of apples by 10%? Can we come up with a formula that gives the desired result of 11 apples?
What if I said I had $10 and I decreased my dollars by 20%? Can we come up with a formula that gives the desired result of $8?
11 10*(1 0.1)
8 10*(1 0.2) *(1 % )New value reference value change
Proof that our two expressions are essentially one in the same
*(1 % )New value reference value change
(1 % )
New valuechange
reference value
1 %
New valuechange
reference value
%
New value reference valuechange
reference value reference value
%
New value reference valuechange
reference value
x y
0 192
1 96
2 48
3 24
Looking at this data set, is there a fixed percent increase or decrease in our y variable?
Exponential Function
For linear functions once we found that the slope stayed the same (that it was a linear function) we then were able to find the y intercept and create a linear model.
Same for exponential functions. Once we know that there is a fixed percent increase or decrease in the y variable, we can then create an exponential model of our data.
An exponential function is just repeating a percent increase or decrease
For example: Let’s say we throw a party and invite 50ppl. They all show up at 7pm, but, since the party is raging, they all invite some of their friends. The party increases by 10% after the first hour. So at 8pm we have:
50*(1 + 0.1) = 55
Exponential Function
At 7pm we had 50ppl and by 8pm we have 55ppl.
Let’s say after another hour our party increases an additional 10%, so this time it’s 10% of 55ppl.
At 9pm (after 2 hours) we’ll have:55 * (1 + 0.1) = 60.5Since 50*(1 + 0.1) = 55, I can write:50 * (1 + 0.1) * (1 + 0.1) = 60.5What if from 9pm to 10pm our 60.5 ppl increases by another 10%?
60.5 * (1 + 0.1) = 66.55 ppl50 * (1 + 0.1) * (1 + 0.1) * (1 + 0.1) =
66.55pplTo recap:50 * (1 + 0.1)0 = 50ppl (after 0 hours)50 * (1 + 0.1)1 = 55ppl (after 1 hour)50 * (1 + 0.1)2 = 60.5ppl (after 2 hours)50 * (1 + 0.1)3 = 66.55ppl (after 3 hours)Can we come up with a formula for the
number of people at the party, where y is the number of people and x is the number of hours after the party has started?
General Exponential Formula
Where y is the new value, P is the initial value, r is the percent increase (+) or decrease (-) in decimal form, and x is the number of repetitions.
How many people will be at the party after 10 hours assuming a 10% increase in people every hour?
*(1 )xy P r 50*(1 0.1)xy
*(1 )xy P r 1050*(1 0.1) 129.7 130y ppl
Two ways to find how many people will be at the party after 10 hours. First is with the formula
Second is with excel…