1. Calculate the population within a 7-mile radius of the city center if the radial population density is
ρ(r) = 5(5 + r2)1/3
(in thousands per square mile). (Round your answer to two decimal places.)
2. A population of insects increases at a rate
280 + 8t + 1.5t2
insects per day. Find the insect population after 5 days, assuming that there are 40 insects at
t = 0.
(Round your answer to the nearest insect.)
consider the population as a collection of rings of thickness dr. Each ring's population is the area of the ring times its density. Add them all up and you get
p(r) = ∫[0,7] 2πr*5(5+r^2)^(1/3) dr
for the insects,
dp/dt = 280 + 8t + 1.5 t^2
p(t) = 40 + 280t + 4t^2 + 3t^3
p(t) = 40 + 280t + 4t^2 + 3t^3
I think it is
p(t) = 40 + 280t + 4t^2 + (1/2)t^3
oh yeah - Good catch
To calculate the population within a 7-mile radius of the city center, we need to integrate the given radial population density function over the area of a circle with a radius of 7 miles.
1. First, let's calculate the area of the circle with a radius of 7 miles:
Area = π * (radius)²
Area = π * (7)²
Area = 49π square miles
2. Now, we need to integrate the radial population density function ρ(r) = 5(5 + r²)^(1/3) over the area we calculated:
Population = ∫[0 to 7] ρ(r) * (Area) dr
However, the given equation represents the population density in thousands per square mile, so we need to multiply the result by 1000 to convert it to actual population.
3. Calculate the integral using the given equation:
Population = 1000 * ∫[0 to 7] 5(5 + r²)^(1/3) * (49π) dr
This integral can be quite complex, and the exact solution involves special functions. Thus, I recommend using a numerical integration method like the Trapezoidal Rule or Simpson's Rule to approximate the integral.
Once you approximate the integral, round the result to two decimal places to find the estimated population within the 7-mile radius.
For the second question, to find the insect population after 5 days, we need to integrate the given rate of increase function over the interval from t = 0 to t = 5.
1. The given rate of increase function is 280 + 8t + 1.5t² insects per day.
2. To find the insect population after 5 days, we need to calculate the definite integral of the rate function:
Population = ∫[0 to 5] (280 + 8t + 1.5t²) dt
3. Integrate the given equation using basic techniques of integration.
Once you calculate the integral, round the result to the nearest insect to find the estimated insect population after 5 days, assuming there are 40 insects at t = 0.