Question 1188652
i think this means that it grows at a constant rate.
for example, something like 10% of its size, no matter what its size is.


the formula for that is f = p * (1 + r) ^ n


f is the future value.
p is the present value
r is the rate of growth per time period.
n is the number of time periods.


they state that it grows from 280 to 1400 in 5 hours.


the formula becomes 1400 = 280 * (1 + r) ^ 5.
divide both sides of this formula by 280 to get:
1400/280 = (1 + r) ^ 5.
take the fifth root of both sides of this equation to get:
(1400/280)^(1/5) = 1 + r.
subtract 1 from both sides of this equation to get:
(1400/280)^(1/5) - 1 = r
solve for r to get:
r = .379729661
that's the rate of growth per hour.


replace r in the original equation and solve for f to get:
f = 280  * (1 + .379729661) ^ 5 = 1400.


the rate is good.
after 4 hours, the population will be:
f = 280 * (1 + .379729661) ^ 4 = 1014.691529.


1014.691529 is your solution to question a.
i'm not sure how they want you to round this so i left it as is.


to find how long it takes to grow to 2630, the formula becomes:


2630 = 280 * (1 + .379729661) ^ n
divide both sides of the equation to 280 and simplify the right side of the equation to get:
2630/280 = 1.379729661 ^ n
take the log of both sides of the equation to get:
log(2630/280) = log(1.379729661 ^ n)
since log(x^n) = n*log(x), this becomes:
log(2630/280) = n * log(1.379729661).
divide both sides of the equation by log(1.379729661) to get:
log(2630/280) / log(1.379729661) = n
solve for n to get:
n = 6.958794448.


confirm by replacing n in the original equation and solving for f to get:
f = 280 * 1.379729661 ^ 6.958794448 = 2630.
round to 2 decimal places to get n = 6.96


the equation can be graphed as shown below:


<img src = "http://theo.x10hosting.com/2021/120912.jpg" >


let me know if you have any questions.


theo