at a given instant the legs of a right triangle are 5cm and 12 cm long. if the short leg is increasing at the rate of 1cm/sec and the long leg is decreasing at the rate of 2cm/sce, how fast is the area changing

let the short leg be x cm

the longer leg be y cm

given: dx/dt = 1 cm/s, dy/dt = -2 cm/s

Area = (1/2)xy
d(Area)/dt = 1/2[x dy/dt + y dx/dt]

at the given instance, x=5,y=12
d(Area)/dt = 1/2[5(-2) + 12(1)]
= 1 cm^2/s

To find how fast the area is changing, we can use the formula for the area of a right triangle: A = 0.5 * base * height.

Given:
- The short leg (base) is increasing at a rate of 1 cm/s.
- The long leg (height) is decreasing at a rate of 2 cm/s.
- The short leg has an initial length of 5 cm, and the long leg has an initial length of 12 cm.

We can differentiate the area formula with respect to time (t) to find how fast the area is changing (dA/dt):

dA/dt = 0.5 * (d(base)/dt) * height + 0.5 * base * (d(height)/dt)

Substituting the known rates of change:

dA/dt = 0.5 * (1 cm/s) * 12 cm + 0.5 * 5 cm * (-2 cm/s)

Simplifying:

dA/dt = 6 cm²/s - 5 cm²/s

dA/dt = 1 cm²/s

Therefore, the area is changing at a rate of 1 cm²/s.

To find how fast the area is changing, we need to differentiate the area formula with respect to time and then substitute the given values.

The area of a right triangle can be calculated using the formula:

Area = (1/2) * Base * Height

In this case, the base is the short leg and the height is the long leg. Let's denote the base as b (which is increasing) and the height as h (which is decreasing).

Given:
b = 5 cm (short leg)
h = 12 cm (long leg)
db/dt = 1 cm/s (rate of change of the short leg)
dh/dt = -2 cm/s (rate of change of the long leg)

Differentiating the area formula with respect to time, we have:

d(Area)/dt = (1/2) * (db/dt * h + b * dh/dt)

Substituting the given values:

d(Area)/dt = (1/2) * (1 cm/s * 12 cm + 5 cm * -2 cm/s)

Simplifying:

d(Area)/dt = (1/2) * (12 cm/s - 10 cm/s)

d(Area)/dt = (1/2) * 2 cm/s

Finally:

d(Area)/dt = 1 cm/s

Therefore, the area is changing at a rate of 1 cm² per second.