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Problem 30 Hard Difficulty

It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species $ S $ of bats living in caves in central Mexico has been related to the surface area $ A $ of the caves by the equation $ S = 0.7A^ {0.3} $.

(a) The cave called Mision Imposible near Puebla, Mexico, has a surface area of $ A = 60 m^2 $ . How many species of bats would you expect to find in that cave?
(b) If you discover that four species of bats live in a cave, estimate the area of the cave.

Answer

a. $S=2.39$
b. $A=332.75 m^{2}$

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Video Transcript

in this problem. We have a relationship between the number of species of bats on the surface area of the cave and for party were given a surface area of 60 square meters and we want to find the number of of species of bats there. So we're going to substitute 60 into the equation for a S equals 0.7 times 60 raised to the 0.3 power. And then we'll put that into a calculator and we get approximately 2.39 species and then in part B suppose there are four species in a cave. What's the surface area? So we're going to substitute for into the equation for s and solve for a We'll start by dividing both sides by 0.7. Now, I'm not going around anything just yet. I'm going to leave everything exact, and the next thing I'm going to do is raise both sides of the equation Toothy power of 1/0 0.3 that will clear out the 0.3 power on the other side. Now, if we put this into the calculator and keep everything exact and don't round until the very end, we get an area of approximately 333 0.58 square meters. However, if you instead round your four divided by 40.7 before you raised to the power that was given in the problem, you are going to get approximately 332 0.75 square meters. So rounding first before you do more does decrease your accuracy, but for both of these answers were in the ballpark of about 333 square meters.

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