Let's solve this step-by-step.

Problem Breakdown:

We are given:

1. The altitude $h$ of the triangle is increasing at a rate of $\frac{dh}{dt} = 1 \, \text{cm/min}$.
2. The area $A$ of the triangle is increasing at a rate of $\frac{dA}{dt} = 1.5 \, \text{cm}^2/\text{min}$.
3. At the moment we are considering, the altitude $h = 7 \, \text{cm}$ and the area $A = 96 \, \text{cm}^2$.

We need to find: The rate at which the base $b$ of the triangle is changing, i.e., $\frac{db}{dt}$.

Key Formula:

The area of a triangle is given by: $A = \frac{1}{2} b h$

Differentiate both sides with respect to time $t$: $\frac{dA}{dt} = \frac{1}{2} \left( b \frac{dh}{dt} + h \frac{db}{dt} \right)$

Solve for $\frac{db}{dt}$:

Rearrange the equation to isolate $\frac{db}{dt}$: $\frac{db}{dt} = \frac{2}{h} \frac{dA}{dt} - \frac{b}{h} \frac{dh}{dt}$

Step 1: Find $b$

From the area formula, $A = \frac{1}{2} b h$, solve for $b$: $b = \frac{2A}{h}$ Substitute $A = 96 \, \text{cm}^2$ and $h = 7 \, \text{cm}$: $b = \frac{2(96)}{7} = \frac{192}{7} \, \text{cm} \approx 27.43 \, \text{cm}.$

Step 2: Substitute values into the differentiated equation

Now substitute all known values into the differentiated formula: $\frac{db}{dt} = \frac{2}{h} \frac{dA}{dt} - \frac{b}{h} \frac{dh}{dt}.$ Substitute $h = 7$, $\frac{dA}{dt} = 1.5$, $\frac{dh}{dt} = 1$, and $b \approx 27.43$: $\frac{db}{dt} = \frac{2}{7}(1.5) - \frac{27.43}{7}(1).$

Simplify: $\frac{db}{dt} = \frac{3}{7} - \frac{27.43}{7}.$ $\frac{db}{dt} = \frac{-24.43}{7} \approx -3.49 \, \text{cm/min}.$

Final Answer:

The base of the triangle is decreasing at a rate of approximately: $\boxed{-3.49 \, \text{cm/min}}.$

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. How does the rate of change in the area depend on both the base and height of the triangle?
2. What happens to the base's rate of change if the altitude is constant?
3. Can this method be extended to other geometric shapes?
4. How would you handle this problem if the base and height were functions of time?
5. What if the rate of change of the area was negative? How would the interpretation differ?

Tip: Always identify known rates and variables before starting the differentiation process in related rates problems.