Question

A triangle has base & centimeters and height h centimeters, where the height is three times the base. Both b and h are functions of time t, measured in seconds. If A represents the area of the triangle, which of the following gives the rate of change of A with respect to t?

A. dA/dt =3bcm/sec
B. dA/dt =2b db/dt cm²/sec
C. dA/dt =3b db/dt cm/sec
D. dA/dt =3b db/dt cm²/sec

Answer 1

The rate of change of the area of a triangle with respect to time cannot be determined without knowing the relationship between the base and time.

Step-by-step explanation:

The formula for the area of a triangle is given by the equation A = 1/2 * base * height. In this case, the base is represented by the variable b and the height is represented by the variable h. Since the height is three times the base, we can write h = 3b. Now, we want to find the rate of change of the area with respect to time, so we need to find dA/dt. To do this, we can use the chain rule of differentiation. We have:

1. dA/dt = (dA/db) * (db/dt)
2. From the formula for the area of a triangle, we can find that dA/db = 1/2 * h, since the base is constant and the height is a function of time. Plugging in h = 3b, we get dA/db = 3/2 * b.
3. To find db/dt, we need to differentiate the function b with respect to t. Since b is a function of t, we can take the derivative of b with respect to t. However, no information is given about the relationship between b and t, so we cannot determine db/dt.

Therefore, the correct answer is None of the above, as we cannot determine the rate of change of A with respect to t without knowing the relationship between b and t.