Odzala National Park in the Congo has a high density of gorillas. Suppose that the radial population density is p(r)=52(1+r^2)^-2 gorillas per square kilometer, where r is the distance from a large grassy clearing with a source of food and water. Calculate the number of gorillas within a 5-km radius of the clearing.
50 pi gorillas
Well, well, well! We've got some gorilla math going on here, don't we? Brace yourself for some "ape-solutely" hilarious calculations!
To find the number of gorillas within a 5-km radius of the clearing, we'll have to integrate the population density function over the given range.
But, before we dive into this gorilla-filled adventure, let's recall that the integral of a function represents the area under the curve. So integrating the population density function will give us the total number of gorillas within a given radius.
So, grab a banana and let's get started!
The integral we need to solve is:
∫[0 to 5] 52(1 + r^2)^-2 dr
Now, let me put on my jester hat and use my hilarious math skills to make this equation my clownish playground.
First, we take the integral of the expression:
∫[0 to 5] 52(1 + r^2)^-2 dr
= 52 ∫[0 to 5] (1 + r^2)^-2 dr
Hmm, things are getting a little wild here. But, don't worry, I've got just the trick up my sleeve!
Using a substitution of u = 1 + r^2, we can rewrite the equation and give ourselves a little more room to play around:
= 52 ∫[u(0)=1 to u(5)=26] u^-2 dr
Now we can integrate:
= 52 ∫[1 to 26] u^-2 du
And what's the integral of u^-2? Let's find out!
= 52 [u^-1 / -1] [1 to 26]
= 52 [(-1/1) - (-1/26)]
= 52 [26 - 1/26]
= 52 [676/26 - 1/26]
= 52 * 675/26
= 1350
Ta-da! After all that clowning around with math, we finally have our answer.
So, within a 5-km radius of the clearing, there are approximately 1350 gorillas roaming about. I hope you found this math journey as entertaining as I did!
To calculate the number of gorillas within a 5-km radius of the clearing, we need to find the area within the 5-km radius and then integrate the given population density function over that area.
The given population density function is p(r) = 52(1+r^2)^-2 gorillas per square kilometer.
To find the area within the 5-km radius, we can use the formula for the area of a circle: A = πr^2, where r is the radius.
Plugging in the value of 5 for r, we get: A = π(5^2) = 25π square kilometers.
Now, we need to integrate the population density function p(r) over the area of 25π square kilometers.
The integral of p(r) over the given area can be written as:
∫[0 to 5] 52(1+r^2)^-2 dr
To solve this integral, it's helpful to use a substitution. Let's substitute u = 1 + r^2, then we can calculate du = 2r dr.
The new integral becomes:
∫[0 to 5] 52(u)^-2 (1/2) du
Simplifying further:
26 ∫[0 to 5] u^-2 du
To integrate u^-2, we can use the power rule. The integral becomes:
26 [-u^(-1)] [0 to 5]
Applying the limits:
26 [-(1/5) - (-(1/0))] = 26 [-(1/5) + (1/0)]
However, division by zero is undefined, so the second term is disregarded.
Therefore, the number of gorillas within a 5-km radius of the clearing is equal to:
26 [-(1/5)] = -26/5 = -5.2 gorillas
It's important to note that obtaining a negative value for the number of gorillas suggests a flaw in the given population density function or the calculations. It's unlikely to have a negative population, so double-checking the function or the boundaries is recommended.