Question: A rectangle has the length 7 times greater than its width. If the rate of change of the area is 28 m2/sec, find the rate of change of the width when the width is 7 m? Find y' derivative
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. **Simplify and 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:
$$\boxed{\frac{dw}{dt} = \frac{2}{7} \text{ m/sec}}$$