The population of a city has been decreasing exponentially since 1990. In 1990, the population was 1,000,000. In 2010, the population was 560,000. If t represents time in years since 1990, which of the following equations best models the decay of the population?
1.
y = 1,000,000(0.98)t
2.
y = 560,000(0.98)t
3.
y = 1,000,000(1.02)t
4.
y = 560,000(1.02)t
think !
1. It starts in 1990 at 1,000,000
2. it DECREASES
and you meant to the POWER t, not times t
a k^t not a k t
to check
at 2010, t = 20 years
y = 1,000,000* .98^20 ?
To determine which equation best models the decay of the population, we need to understand the concept of exponential decay and how it relates to the given data.
Exponential decay is a mathematical function that describes a situation where a quantity decreases exponentially over time. The general form of an exponential decay function is: y = a * (b)^t, where y represents the final value, a represents the initial value, b represents the decay factor, and t represents the time.
In this case, the initial population in 1990 is given as 1,000,000. This means that the value of a in the function should be 1,000,000. The population in 2010 is given as 560,000, which means the final value, y, should be 560,000.
Now let's analyze each equation:
1. y = 1,000,000(0.98)^t
2. y = 560,000(0.98)^t
3. y = 1,000,000(1.02)^t
4. y = 560,000(1.02)^t
Equation 1 and Equation 3 have opposite signs on the exponent. This indicates growth rather than decay, so we can eliminate them.
We are left with Equation 2 and Equation 4. To determine the correct equation, we can compare the initial and final population values given in the question.
In the equation y = 560,000(0.98)^t, the initial value should be 560,000, but this contradicts the given initial value of 1,000,000. Therefore, we can eliminate Equation 2.
The correct equation that best models the decay of the population is Equation 4:
y = 560,000(1.02)^t
In this equation, the initial value of 560,000 matches the given value, and the positive exponent on 1.02 ensures exponential decay over time.
So, Equation 4 is the correct answer.
First of all looking at your choices, only 1. and 2. are logical, the other two would show an increase not a decrease, so ....
let the rate be r
560000 = 1000000 r^10
.56 = r^10
take 10th root of both sides using your calculator
.56^(1/10) = r
r = .94367..
so y = 560000(.94)^t would be it using 2 decimals like all their answers
I don't see that choice!!!
check:
.94367^10 = .560017...
.560017...(1000000) = 560,017 close enough
My answer is correct