The base of a triangle is increasing in length at a rate of 3cm per second, while its height is decreasing at a rate of 2cm per second. At the moment when the area of the triangle is 60cm2, its base is 20cm long. Determine the rate of change of the area at this time.
Q. The base of a triangle is increasing in length at a rate of 3cm per second, while its height is decreasing at a rate of 2cm per second. At the moment when the area of the triangle is 60cm2, its base is 20cm long. Determine the rate of change of the area at this time.
Area Formula Derivation: We know that the area of a triangle is given by the formula A=21×base×height. Let's denote the base as b and the height as h. We are given that the base b is increasing at a rate of dtdb=3cm/s and the height h is decreasing at a rate of dtdh=−2cm/s (negative because it's decreasing). We need to find the rate of change of the area dtdA when the area A is 60cm2 and the base b is b1.
Height Calculation: First, let's find the height h of the triangle when the area A is 60 cm2 and the base b is 20 cm. Using the area formula A=21×b×h, we can solve for h:60=21×20×hh=(21×20)60h=1060h=6 cmSo, the height of the triangle at this moment is 6 cm.
Rate of Change Calculation: Now, let's use the formula for the area of a triangle to find the rate of change of the area with respect to time. We differentiate both sides of the equation A=21⋅b⋅h with respect to time t:dtdA=21⋅(dtdb⋅h+b⋅dtdh)We know dtdb=3 cm/s, dtdh=−2 cm/s, b=20 cm, and h=6 cm. Let's plug these values into the equation:dtdA=21⋅(3⋅6+20⋅−2)
Final Result: Now we perform the calculations:dtdA=21×(18−40)dtdA=21×(−22)dtdA=−11cm2/sThe negative sign indicates that the area of the triangle is decreasing at a rate of 11cm2 per second at the moment when the base is 20cm and the area is 60cm2.