In 1983, the population of Functionville was 2560, and it tripled in 6 years.
a. Write an exponential model for the population of Functionville t years after 1983.
b. According to your model, it what year had the population grown to 7 times its size?
since it tripled in 6 years, ew have
P = 2560*3^(t/6) where t=years since 1983
now plug in P=7, get t, add 1983 and voila'
ahem. Plug in P=7*2560
I'm on the same problem and I'm having trouble solving it, can you go into a bit more detail.
I'm confused
a. To write an exponential model for the population of Functionville, we can use the formula: P(t) = P0 * r^t, where P(t) is the population at time 't', P0 is the initial population, and r is the growth rate.
In this case, the initial population in 1983 is 2560, and we know that it tripled in 6 years. To find the growth rate, we can divide the final population by the initial population and then take the 6th root to account for the 6-year period:
Growth rate (r) = (Final population / Initial population) ^ (1 / t)
= (3 * Initial population) ^ (1 / t)
= (3 * 2560) ^ (1 / 6)
= 480 ^ (1 / 6)
Therefore, the exponential model for the population of Functionville is:
P(t) = 2560 * (480 ^ (t / 6))
b. According to the exponential model, we need to find the value of 't' when the population has grown to 7 times its size. Let's substitute this value into the equation and solve for 't':
7 * Initial population = P0 * r^t
7 * 2560 = 2560 * (480 ^ (t / 6))
Now we can simplify the equation:
7 = 480 ^ (t / 6)
To solve for 't', we need to take the logarithm of both sides of the equation:
log(7) = log(480 ^ (t / 6))
Using the property of logarithms, we can bring the exponent down:
log(7) = (t / 6) * log(480)
Now we can isolate 't' by multiplying both sides by 6 and dividing by log(480):
t = (6 * log(7)) / log(480)
Using a calculator, we can find the approximate value of 't'.