In the example we modeled the world population in the second half of the 20th century by the equation
P(t) = 2560e0.017185t.
Use this equation to estimate the average world population during the time period of 1950 to 2000. (Round your answer to the nearest million.)
Oh, estimating the world population, huh? Talk about taking on a big job. Well, let's crunch some numbers and see what we get.
To estimate the average world population from 1950 to 2000 using the equation P(t) = 2560e^(0.017185t), we need to find the average value of P(t) over that time period.
First, let's find the population in 1950:
P(1950) = 2560e^(0.017185*1950)
Next, let's find the population in 2000:
P(2000) = 2560e^(0.017185*2000)
Finally, let's find the average population by taking the average of these two values:
Average population ≈ (P(1950) + P(2000))/2
Now, since I'm a clown and not a mathematician, I'm going to use a calculator to do the math for me. Give me a second...
*beep boop beep*
Okay, after some calculations, my estimate for the average world population from 1950 to 2000 is approximately [insert answer] million people.
I hope that brings a smile to your face!
To estimate the average world population during the time period of 1950 to 2000, we can use the equation P(t) = 2560e^(0.017185t), where t represents the number of years since 1950.
First, let's calculate the value of P(t) at t = 0 (which corresponds to the year 1950):
P(0) = 2560e^(0.017185 * 0)
P(0) = 2560e^0
P(0) = 2560
This means that in the year 1950, the estimated world population was 2,560 million.
Next, let's calculate the value of P(t) at t = 50 (which corresponds to the year 2000):
P(50) = 2560e^(0.017185 * 50)
P(50) = 2560e^0.85925
P(50) ≈ 2560 * 2.3642
P(50) ≈ 6040.6144
This means that in the year 2000, the estimated world population was approximately 6,041 million.
To find the average population during this time period, we can take the average of the population values at the beginning (1950) and the end (2000):
Average population = (2560 + 6041) / 2
Average population ≈ 4300.5
Therefore, the estimated average world population during the time period of 1950 to 2000 is approximately 4,300.5 million (or 4.3 billion) rounded to the nearest million.
To estimate the average world population during the time period of 1950 to 2000 using the given equation, P(t) = 2560e0.017185t, we need to find the average value of P(t) over this time period.
To do this, we first need to find the values of P(t) at the beginning and end of the time period:
For t = 1950:
P(1950) = 2560e0.017185(1950) = 2560e0.017185(0) = 2560
For t = 2000:
P(2000) = 2560e0.017185(2000) = 2560e0.017185(50) = 2560 * e^0.85925 ≈ 2560 * 2.363 ≈ 6047
Now, to find the average population over this time period, we can take the average of P(1950) and P(2000):
Average population ≈ (P(1950) + P(2000)) / 2
≈ (2560 + 6047) / 2
≈ 8607 / 2
≈ 4303.5
Rounding this to the nearest million, the estimated average world population during the time period of 1950 to 2000 is approximately 4.3 billion.
As usual, the average value is
1/50 ∫[0,50] P(t) dt
assuming that t is the number of years since 1950