Answer:
49 cm²/s
Step-by-step explanation:
The problem statement tells you what to do.
Write an equation relating A, b, h:
A = bh . . . . . . the equation for the area of a rectangle
Differentiate with respect to t:
dA/dt = (db/dt)h + b(dh/dt) . . . . . . . product rule
To find the rate of change after 18 seconds, you need to know the dimensions b and h after 18 seconds. Since each dimension was increasing at the rate of 1 cm/s, it is 18 cm more than it was at the beginning:
At 18 seconds,
b = 6 cm + 18 cm = 24 cm;
h = 7 cm + 18 cm = 25 cm.
Of course, db/dt = dh/dt = 1 cm/s. Then the rate of change of area is ...
dA/dt = (1 cm/s)(25 cm) + (24 cm)(1 cm/s)
dA/dt = 49 cm²/s
_____
You could write a formula for the area as a function of time and differentiate that:
A = (6 +t)(7 +t) = 42 + 13t + t²
Then the derivative is ...
dA/dt = 13 +2t
and when t=18, this is ...
dA/dt = 13 + 2(18) = 49 . . . . cm²/s
