The base of a triangle is shrinking at a rate or 2 cm/s while the height is growing at a rate of 3 cm/s. How quickly is the area of the triangle changing when the base is 10 cm and the height is 6 cm?
To find how quickly the area of the triangle is changing, we can use the formula for the area of a triangle: A = (1/2) * base * height.
Given that the base is shrinking at a rate of 2 cm/s, we'll use a negative sign to represent this decrease. We also know that the height is growing at a rate of 3 cm/s, so we'll use a positive sign to represent this increase.
Let's differentiate the formula for the area of a triangle with respect to time (t) using the chain rule:
dA/dt = (1/2) * (d(base)/dt) * height + (1/2) * base * (d(height)/dt)
Now let's substitute the given values when the base is 10 cm and the height is 6 cm:
base = 10 cm
height = 6 cm
(d(base)/dt) = -2 cm/s (since it is shrinking at a rate of 2 cm/s)
(d(height)/dt) = 3 cm/s (since it is growing at a rate of 3 cm/s)
Substituting these values into the formula, we have:
dA/dt = (1/2) * (-2 cm/s) * 6 cm + (1/2) * 10 cm * (3 cm/s)
Now we can calculate the result:
dA/dt = -6 cm^2/s + 15 cm^2/s
dA/dt = 9 cm^2/s
Therefore, the area of the triangle is changing at a rate of 9 cm^2/s when the base is 10 cm and the height is 6 cm.