The altitude of a triangle is increasing at a rate of 1.500 centimeters/minute while the area of the triangle is increasing at a rate of 2.000 square centimeters/minute. At what rate is the base of the triangle changing when the altitude is 11.000 centimeters and the area is 89.000 square centimeters?
To find the rate at which the base of the triangle is changing, we need to use the formula for the area of a triangle.
The area (A) of a triangle can be calculated using the formula A = (1/2) * base * altitude.
Let's denote the base of the triangle as 'x' centimeters. The altitude is given as 11.000 centimeters, and the area is given as 89.000 square centimeters.
Differentiating both sides of the area formula with respect to time (t), we get
dA/dt = (1/2) * (dx/dt) * altitude + (1/2) * base * (d(altitude)/dt)
We are given dA/dt (the rate of change of area) as 2.000 square centimeters/minute, and d(altitude)/dt (the rate of change of altitude) as 1.500 centimeters/minute. We need to find dx/dt (the rate of change of the base) when altitude is 11.000 centimeters and area is 89.000 square centimeters.
Substituting the given values into the formula, we get
2.000 = (1/2) * (dx/dt) * 11.000 + (1/2) * x * 1.500
2.000 = (11/2) * (dx/dt) + (3/2) * x
Now, we can solve this equation to find dx/dt.
2.000 - (3/2) * x = (11/2) * (dx/dt)
dx/dt = (2.000 - (3/2) * x) / (11/2)
Simplifying further,
dx/dt = (4.000 - 3x) / 11
Now we can substitute the values of x = base, altitude = 11.000, and solve for dx/dt.
dx/dt = (4.000 - 3 * 11.000) / 11
dx/dt = (4.000 - 33.000) / 11
dx/dt = -29.000 / 11
dx/dt = -2.636 centimeters/minute
Therefore, the base of the triangle is changing at a rate of -2.636 centimeters/minute when the altitude is 11.000 centimeters and the area is 89.000 square centimeters.