Questions: For many species of fish, the weight W is a function of the length x, given by W=kx^3, where k is a constant depending on the species. Suppose k=0.008, W is in pounds, and x is in inches, the weight is W(x)=0.008x^3 a. Find the inverse function of this function b. What does the inverse function give? c. Use the inverse function to find the length of a fish that weighs 216 pounds d. In the context of the application, what are the domain and range of the inverse function? a. Give the inverse function W^(-1)(x)=∛(125x) b. What does the inverse function give? Choose the correct answer below. A. Given the weight, the inverse function calculates the length B. Given the length, the fiverse function calculates the species constant, k C. Given the length, the inverse function calculates the weight D. Given the weight, the inverse function calculates the species constant, k

For many species of fish, the weight W is a function of the length x, given by W=kx^3, where k is a constant depending on the species. Suppose k=0.008, W is in pounds, and x is in inches, the weight is W(x)=0.008x^3
a. Find the inverse function of this function
b. What does the inverse function give?
c. Use the inverse function to find the length of a fish that weighs 216 pounds
d. In the context of the application, what are the domain and range of the inverse function?
a. Give the inverse function
W^(-1)(x)=∛(125x)
b. What does the inverse function give? Choose the correct answer below.
A. Given the weight, the inverse function calculates the length
B. Given the length, the fiverse function calculates the species constant, k
C. Given the length, the inverse function calculates the weight
D. Given the weight, the inverse function calculates the species constant, k

Transcript text: For many species of fish, the weight W is a function of the length x , given by $\mathrm{W}=\mathrm{k} \mathrm{x}^{3}$, where k is a constant depending on the species. Suppose $\mathrm{k}=0.008, \mathrm{~W}$ is in pounds, and x is in inches, the weight is $\mathrm{W}(\mathrm{x})=0.008 \mathrm{x}^{3}$ a. Find the inverse function of this function b. What does the inverse function give? c. Use the inverse function to find the length of a fish that weighs 216 pounds d. In the context of the application, what are the domain and range of the inverse function? a. Give the inverse function $W^{-1}(x)=\sqrt[3]{125 x}$ b. What does the inverse function give? Choose the correct answer below. A. Given the weight, the inverse function calculates the length B. Given the length, the fiverse function calculates the species constant, $k$ C. Given the length, the inverse function calculates the weight D. Given the weight, the inverse function calculates the species constant, k

Solution

Solution Steps

To solve the given problem, we need to find the inverse of the function $ W(x) = 0.008x^3 $. The inverse function will allow us to determine the length of the fish given its weight. We will then use this inverse function to find the length of a fish that weighs 216 pounds.

a. To find the inverse function, solve the equation $ W = 0.008x^3 $ for $ x $.

b. The inverse function provides the length of the fish when given its weight.

c. Use the inverse function to calculate the length of a fish that weighs 216 pounds.

Step 1: Find the Inverse Function

To find the inverse function of $ W(x) = 0.008x^3 $, we start by setting $ W = 0.008x^3 $ and solving for $ x $:

\[ x = \left( \frac{W}{0.008} \right)^{\frac{1}{3}} \]

Thus, the inverse function is given by:

\[ W^{-1}(W) = \left( \frac{W}{0.008} \right)^{\frac{1}{3}} \]

Step 2: Calculate the Length for a Given Weight

Next, we use the inverse function to find the length of a fish that weighs $ W = 216 $ pounds:

\[ L = W^{-1}(216) = \left( \frac{216}{0.008} \right)^{\frac{1}{3}} \]

Calculating this gives: