Given the problem's context and your question about the rate at which the area of the rectangle is increasing, let's analyze it using related rates.

Given information:

• The length $l$ of the rectangle is increasing at a rate of $\frac{dl}{dt} = 8 \, \text{cm/s}$.
• The width $w$ of the rectangle is increasing at a rate of $\frac{dw}{dt} = 3 \, \text{cm/s}$.
• When the length is $l = 10 \, \text{cm}$ and the width is $w = 6 \, \text{cm}$, we need to find how fast the area $A$ of the rectangle is increasing.

Step 1: Differentiate the area formula

The area of a rectangle is given by: $A = l \cdot w$

To find how fast the area is increasing, we need to differentiate $A$ with respect to time $t$: $\frac{dA}{dt} = \frac{\partial A}{\partial l} \cdot \frac{dl}{dt} + \frac{\partial A}{\partial w} \cdot \frac{dw}{dt}$

Since $\frac{\partial A}{\partial l} = w$ and $\frac{\partial A}{\partial w} = l$, the equation becomes: $\frac{dA}{dt} = w \cdot \frac{dl}{dt} + l \cdot \frac{dw}{dt}$

Step 2: Substitute the values

Substituting the given values:

• $l = 10 \, \text{cm}$
• $w = 6 \, \text{cm}$
• $\frac{dl}{dt} = 8 \, \text{cm/s}$
• $\frac{dw}{dt} = 3 \, \text{cm/s}$

$\frac{dA}{dt} = 6 \cdot 8 + 10 \cdot 3$ $\frac{dA}{dt} = 48 + 30$ $\frac{dA}{dt} = 78 \, \text{cm}^2/\text{s}$

Conclusion

The area of the rectangle is increasing at a rate of $78 \, \text{cm}^2/\text{s}$.

Would you like more details or have any questions about this calculation?

Here are five related questions to further explore this topic:

1. What happens to the rate of area change if the width's rate of increase doubles?
2. How does the formula change if only the length of the rectangle increases while the width stays constant?
3. What is the geometric interpretation of related rates in this context?
4. How can we apply this method to other shapes, like triangles or circles?
5. What if the length and width were decreasing instead of increasing—how would that affect the rate of area change?

Tip: In related rates problems, it's helpful to differentiate all quantities with respect to time before substituting values to ensure accuracy.