Question \( 91,4.8 .141-\mathrm{Ce} \) Allometric relations often can be modeled by \( f(x)=a x^{b} \), 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 \( f(x)=0.445 x^{1.22} \). 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 3-gram crab. 1.25 of 92 points (b) Approximate the weight of a crab that has \( 0.9- \) gram claws. (a) The claws of a 3 -gram crab weigh approximately \( \square \) grams. (Type an integer or decimal rounded to two decimal places as needed.)

Let \( f(x)=0.445 x^{1.22} \) and note that \( x \) is the crab’s weight in grams. **(a) For a 3-gram crab:** We compute \[ f(3)=0.445 \cdot 3^{1.22}. \] 1. Compute \( 3^{1.22} \): \[ 3^{1.22}=\exp(1.22\ln(3)). \] Using \( \ln(3) \approx 1.0986 \), we have: \[ 1.22\ln(3) \approx 1.22 \cdot 1.0986 \approx 1.3413, \] and hence: \[ 3^{1.22} \approx \exp(1.3413) \approx 3.822. \] 2. Multiply by \( 0.445 \): \[ f(3) \approx 0.445 \cdot 3.822 \approx 1.699. \] Rounded to two decimal places, the claws weigh approximately \( 1.70 \) grams. **(b) For claws weighing \( 0.90 \) grams:** We set \[ f(x)=0.445 x^{1.22}=0.90. \] Solve for \( x \): 1. Divide both sides by \( 0.445 \): \[ x^{1.22}=\frac{0.90}{0.445} \approx 2.0225. \] 2. Take the \( 1.22 \)th root (or raise both sides to the power \( \frac{1}{1.22} \)): \[ x=(2.0225)^{\frac{1}{1.22}}. \] Taking natural logarithms: \[ \ln(x)=\frac{1}{1.22}\ln(2.0225). \] With \( \ln(2.0225) \approx 0.7038 \), we have: \[ \ln(x) \approx \frac{0.7038}{1.22} \approx 0.5769, \] and thus: \[ x\approx\exp(0.5769) \approx 1.78. \] Rounded to two decimal places, the crab weighs approximately \( 1.78 \) grams. **Final Answers:** (a) The claws of a 3-gram crab weigh approximately \( 1.70 \) grams. (b) A crab with \( 0.90 \)-gram claws weighs approximately \( 1.78 \) grams.