Allometric relations often can be modeled by f(x)= x^b, where a and b are constants. One study showed that for a male fiddler crab weighing over 0.75 gram, the weight of its claws can be estimated by f(x)= 0.448 x^1.21. The input x is the weight of the crab in grams, and the output f(x) is the weight of the claws in grams.

(a) Predict the weight of the claws of a 2-gram crab

(b) Approximate the weight of a crab that has 0.5-gram claws

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dfbustos dfbustos · Answer 1 · 2020-05-18T03:41:52+02:00

Answer:

a) [tex] f(2)= 0.448 (2)^{1.21}= 1.036 grams[/tex]

b) [tex] 0.5 = 0.448 x^{1.21}[/tex]

Dividing both sides by 0.448 we got:

[tex]\frac{0.5}{0.448} = x^{1.21}[/tex]

We can appy the exponent [tex]\frac{1}{1.21}[/tex] in both sides of the equation and we got:

[tex] (\frac{0.5}{0.448})^{\frac{1}{1.21}} = x= 1.095grams[/tex]

Step-by-step explanation:

For this case we know the following function:

[tex] f(x) = 0.448 x^{1.21}[/tex]

The notation is: x is the weight of the crab in grams, and the output f(x) is the weight of the claws in grams.

Part a

For this case we just need to replace x = 2 gram in the function and we got:

[tex] f(2)= 0.448 (2)^{1.21}= 1.036 grams[/tex]

Part b

For this case we know tha value for [tex] f(x) =0.5[/tex] and we want to find the value of x who satisfy this condition:

[tex] 0.5 = 0.448 x^{1.21}[/tex]

Dividing both sides by 0.448 we got:

[tex]\frac{0.5}{0.448} = x^{1.21}[/tex]

We can appy the exponent [tex]\frac{1}{1.21}[/tex] in both sides of the equation and we got:

[tex] (\frac{0.5}{0.448})^{\frac{1}{1.21}} = x= 1.095grams[/tex]