1. **State the problem:** We have a man 5 feet 7 inches tall losing weight at 5 pounds per month. The relationship between body mass index $B$ and weight $w$ (in pounds) is given by: $$4600B = 710w$$ We need to find the rate of change of body mass index $\frac{dB}{dt}$ with respect to time. 2. **Rewrite the equation:** $$4600B = 710w$$ 3. **Differentiate both sides with respect to time $t$:** Using implicit differentiation, $$4600 \frac{dB}{dt} = 710 \frac{dw}{dt}$$ 4. **Substitute the known rate of weight change:** Given $\frac{dw}{dt} = -5$ pounds per month (losing weight means negative rate), $$4600 \frac{dB}{dt} = 710 (-5)$$ 5. **Solve for $\frac{dB}{dt}$:** $$\frac{dB}{dt} = \frac{710 \times (-5)}{4600} = \frac{-3550}{4600} = -\frac{355}{460} = -\frac{71}{92} \approx -0.7717$$ 6. **Interpretation:** The body mass index is decreasing at approximately $0.7717$ units per month. **Final answer:** $$\boxed{\frac{dB}{dt} \approx -0.7717 \text{ per month}}$$