Calculating the Rate of Change of Area in a Right Triangle

The problem at hand involves a right triangle ABC with a fixed length of AB and a variable length of BC. The triangle's area changes as the length of BC increases at a constant rate. Let's explore the mathematical concepts and calculations involved in determining the rate of change of the area of this right triangle.

Setting Up the Problem

We have a right triangle ABC with angle B being 90 degrees. The length of AB is fixed at 8 cm, and the length of BC is increasing at a rate of 0.5 cm per second. The area of a right triangle can be calculated using the formula:

Area (1/2) × base × height

In this case, we can consider AB as the height and BC as the base. The height of triangle ABC is a constant 8 cm, and the base BC is increasing over time.

Determining the Rate of Change of Area

The rate of change of the area can be found by differentiating the area formula with respect to time. First, we need to express the area as a function of BC, denoted as t (time), and then differentiate it.

Area (1/2) × 8 × BC 4 × BC

Now, to find the rate of change of the area, we differentiate both sides with respect to time t:

d(Area)/dt 4 × d(BC)/dt

Given that d(BC)/dt 0.5 cm/s, we can substitute this value into the equation:

d(Area)/dt 4 × 0.5 2 cm2/s

This indicates that the area of the triangle is increasing at a rate of 2 square centimeters per second.

Alternative Calculation Method

Let's consider a more intuitive approach to confirm our result. Suppose at time t 0, the length of BC is L cm. After 1 second, the length of BC increases to L 0.5 cm. The area of the triangle at time t 0 is:

Area at t 0 (1/2) × 8 × L 4L cm2

The area of the triangle at t 1s is:

Area at t 1s (1/2) × 8 × (L 0.5) 4L 2 cm2

The increase in area over that one second is:

Increase in area (4L 2) - 4L 2 cm2

This confirms that the area is increasing at a rate of 2 square centimeters per second, regardless of the initial length of BC.

Conclusion

The rate of change of the area of triangle ABC is directly proportional to the rate of increase of the base BC. Given the fixed height of 8 cm and a base increasing at 0.5 cm per second, the area of the triangle increases at a rate of 2 square centimeters per second. This demonstrates the mathematical relationship and the practical implications of changes in geometric dimensions on the area of a right triangle.