Visualize an interesting natural setting with a normal sized beaver, around 20 kilograms, marking its territory in a lush forest, defining the boundaries of its defended region which spreads across a notable 200 square meters. Nearby, carve out an imprint of an ancient, huge beaver of North America, approximately 3.5 meters long, weighting about 200 kg, representing the grandiosity of its defended ground. Fossils offer evidence of these ancient beings. Ensure both beavers are represented in harmony with their surroundings, providing a clear picture of the significant difference in their sizes and the areas they would defend. There should be no text in the image.

An animal will defend the region around its home by attacking intruders that come in to the region. According to the statistical studies reported in J.M. Emlen's Ecology: An Evolutionary Approach, the defended region's area varies directly with the 1.31 power of the animal's body mass.

1. Suppose that a normal 20 kilogram beaver will defend a region of area 200 square meters. Write an equation for this function

2. Skeletons show that thousands of years ago North American beavers were up to 3.5 meters long and had a mass of 200 kg. How many square meters should a beaver have defended?

Area=k*mass^1.31

1) solve for k, using mass, area given.
2) now on the second, use the same k, solve for area.

I am wondering how this is calculus.

its part of a precalc-calc class.

idk?
thanks though

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1. Let's denote the defended region's area as A (in square meters) and the animal's body mass as M (in kilograms). According to the given information, the defended region's area varies directly with the 1.31 power of the animal's body mass. This can be expressed as:

A = k * M^1.31

Where k is the constant of proportionality. To find the equation for this specific case with a normal 20-kilogram beaver defending a region of area 200 square meters, we can substitute the values into the equation:

200 = k * 20^1.31

Now we need to solve for k. Rearranging the equation, we have:

k = 200 / (20^1.31)

Calculating the right side of the equation:

k ≈ 200 / (20^1.31) ≈ 0.1621

Thus, the equation for this specific case is approximately:

A = 0.1621 * M^1.31

2. To determine how many square meters a beaver with a mass of 200 kg should have defended, we can use the equation we derived in question 1. Let M represent the beaver's body mass (200 kg) and A represent the defended region's area we need to find.

A = 0.1621 * M^1.31

Replacing M with 200 kg:

A = 0.1621 * (200^1.31)

Calculating the right side of the equation:

A ≈ 0.1621 * (200^1.31) ≈ 6,426.77

Therefore, a beaver with a mass of 200 kg should have defended an area of approximately 6,426.77 square meters.