1. the altitude of a triangle is increasing at a rate of 2.5 cm/min while the area of the triangle is increasing at a rate of 5 square cm per minute. At what rate is the base of the triangle changing when altitude is 9 cm and the area is 88 square cm

To solve this problem, we need to use related rates. We'll start by expressing the relationship between the area of the triangle, the base, and the altitude.

The area \( A \) of a triangle is given by:
\[ A = \frac{1}{2} \times \text{base} \times \text{altitude} \]

Let \( A \) be the area of the triangle, \( b \) be the base of the triangle, and \( h \) be the altitude of the triangle. Then,
\[ A = \frac{1}{2} b h \]

Given:
- The altitude \( h \) is increasing at a rate of \( \frac{dh}{dt} = 2.5 \) cm/min.
- The area \( A \) is increasing at a rate of \( \frac{dA}{dt} = 5 \) cm²/min.

We are to find the rate at which the base \( b \) is changing, \( \frac{db}{dt} \), when \( h = 9 \) cm and \( A = 88 \) cm².

First, we differentiate the area formula with respect to time \( t \):
\[ \frac{dA}{dt} = \frac{1}{2} \left( b \frac{dh}{dt} + h \frac{db}{dt} \right) \]

We can solve for \( \frac{db}{dt} \):
\[ 5 = \frac{1}{2} \left( b (2.5) + 9 \frac{db}{dt} \right) \]

Simplify and solve for \( \frac{db}{dt} \):
\[ 10 = b (2.5) + 9 \frac{db}{dt} \]

Next, we need to find the value of \( b \) when \( h = 9 \) cm and \( A = 88 \) cm²:
\[ A = \frac{1}{2} b h \]
\[ 88 = \frac{1}{2} b (9) \]
\[ 88 = 4.5b \]
\[ b = \frac{88}{4.5} \]
\[ b \approx 19.56 \, \text{cm} \]

Now, substitute \( b \approx 19.56 \) and \( \frac{dh}{dt} = 2.5 \) into the equation:
\[ 10 = 19.56 (2.5) + 9 \frac{db}{dt} \]
\[ 10 = 48.9 + 9 \frac{db}{dt} \]
\[ 10 - 48.9 = 9 \frac{db}{dt} \]
\[ -38.9 = 9 \frac{db}{dt} \]
\[ \frac{db}{dt} = \frac{-38.9}{9} \]
\[ \frac{db}{dt} \approx -4.32 \, \text{cm/min} \]

Thus, the base of the triangle is decreasing at a rate of approximately 4.32 cm/min.

Would you like further details or have any questions?

Here are some related questions you might be interested in:
1. How do you apply the chain rule in related rates problems?
2. Can you explain how to solve problems involving rates of change in geometry?
3. What are some common applications of related rates in physics?
4. How can you use implicit differentiation to solve related rates problems?
5. What are the steps to set up and solve a related rates problem?

**Tip:** Always make sure to correctly identify the relationships between variables and their rates of change when solving related rates problems.

Learn how to solve a related rates problem involving the change in base of a triangle when given its altitude and area rates. Step-by-step solution using the area formula and differentiation. Suitable for advanced high school students and calculus learners.

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