Rate Width 84675B

1. **State the problem:** We have a rectangle where the length $L$ is 7 times the width $W$, so $L = 7W$. The area $A$ of the rectangle is changing at a rate of $\frac{dA}{dt} = 28$ m$^2$/sec. We need to find the rate of change of the width $\frac{dW}{dt}$ when $W = 7$ m. 2. **Write the formula for the area:** $$A = L \times W$$ Since $L = 7W$, substitute: $$A = 7W \times W = 7W^2$$ 3. **Differentiate both sides with respect to time $t$:** $$\frac{dA}{dt} = 7 \times 2W \frac{dW}{dt} = 14W \frac{dW}{dt}$$ 4. **Plug in the known values:** Given $\frac{dA}{dt} = 28$ and $W = 7$, substitute: $$28 = 14 \times 7 \times \frac{dW}{dt}$$ 5. **Solve for $\frac{dW}{dt}$:** $$28 = 98 \frac{dW}{dt}$$ Divide both sides by 98: $$\frac{28}{\cancel{98}} = \frac{\cancel{98}}{98} \frac{dW}{dt} \Rightarrow \frac{2}{7} = \frac{dW}{dt}$$ 6. **Final answer:** The rate of change of the width when $W=7$ m is: $$\boxed{\frac{dW}{dt} = \frac{2}{7} \text{ m/sec}}$$