The formula B=0.4089M^(3)/(4) gives the bird inhalation rate, B cubic metres of air per day, for a bird with mass M kilograms. a) Rewrite the formula using radicals. b) Calculate the inhalation rate for each bird. i) a 4.5-kg bald eagle ii) a 8.0-kg Canada goose c) Determine the mass of a bird whose inhalation rate is twice that of the bald eagle. d) Is the mass in part c twice that of the bald eagle? Explain.

## Step 1: Rewrite the formula using radicals<br />### The fractional exponent can be rewritten as a radical. $M^{\frac{3}{4}}$ is equivalent to $\sqrt[4]{M^3}$ or $(\sqrt[4]{M})^3$. Thus, the formula becomes $B = 0.4089\sqrt[4]{M^3}$ or $B = 0.4089(\sqrt[4]{M})^3$.<br /><br />## Step 2: Calculate the inhalation rate for a 4.5-kg bald eagle<br />### Substitute $M = 4.5$ into the formula: $B = 0.4089(4.5)^{\frac{3}{4}} \approx 0.4089(2.83) \approx 1.16$. The bald eagle's inhalation rate is approximately $1.16$ $m^3$ of air per day.<br /><br />## Step 3: Calculate the inhalation rate for an 8.0-kg Canada goose<br />### Substitute $M = 8.0$ into the formula: $B = 0.4089(8.0)^{\frac{3}{4}} \approx 0.4089(5.66) \approx 2.31$. The Canada goose's inhalation rate is approximately $2.31$ $m^3$ of air per day.<br /><br />## Step 4: Determine the mass of a bird whose inhalation rate is twice that of the bald eagle<br />### The bald eagle's inhalation rate is approximately $1.16$ $m^3$/day. Twice this rate is $2.32$ $m^3$/day. We need to solve for $M$ in the equation $2.32 = 0.4089M^{\frac{3}{4}}$. Dividing both sides by $0.4089$ gives $M^{\frac{3}{4}} \approx \frac{2.32}{0.4089} \approx 5.67$. Raising both sides to the power of $\frac{4}{3}$ gives $M \approx 5.67^{\frac{4}{3}} \approx 8.0$. The mass of the bird is approximately $8.0$ kg.<br /><br />## Step 5: Determine if the mass in part c is twice that of the bald eagle and explain<br />### The mass of the bald eagle is $4.5$ kg. Twice this mass is $9.0$ kg. The mass of the bird with twice the inhalation rate is $8.0$ kg, which is not twice the mass of the bald eagle. This is because the relationship between the mass and the inhalation rate is not linear, but rather follows a power law with an exponent of $\frac{3}{4}$.