To understand the relationship between the rate at which the altitude of a triangle is increasing and the rate at which the area is increasing, we need to consider the formula for the area of a triangle:
Area = (1/2) × base × height
In this scenario, we are given:
- The altitude (height) of the triangle is increasing at a rate of 1 cm/min.
- The area of the triangle is increasing at a rate of 2 cm²/min.
Let’s denote:
- A: area of the triangle
- b: base of the triangle
- h: height (altitude) of the triangle
- dh/dt: rate of change of height with respect to time
- dA/dt: rate of change of area with respect to time
From the area formula, we can derive the following relationship:
To find the derivative of the area with respect to time, we apply the product rule:
dA/dt = (1/2) × b × (dh/dt)
Now plugging in our known values:
- dA/dt = 2 cm²/min
- dh/dt = 1 cm/min
Substituting these values into the equation gives:
2 = (1/2) × b × 1
From this, we can solve for the base b:
2 = (1/2) × b
=> b = 4 cm
Therefore, the base of the triangle must be 4 cm for the given rates of change to hold true. This demonstrates how the triangle’s dimensions interact, indicating that an increase in height, even at a constant base, results in a proportional increase in the area of the triangle.
In summary, as the altitude increases at a rate of 1 cm/min, the area responds by increasing at a rate of 2 cm²/min, provided the base remains constant at 4 cm.