A rectangle has sides of length x cm and 2x - 4cm and the length x cm at time t
seconds is given by x = 2 + 3t, (t>=0) . Show that the area, A cm^2 , the rectangle, in
terms of t is A =12t + 18t^2 . Hence find the rate of change of the area at the instant when
t = 2.
To find the area of the rectangle in terms of t, we need to use the formula for the area of a rectangle, which is A = length * width.
In this case, the length of the rectangle is x cm and the width is 2x - 4 cm.
Given that x = 2 + 3t, we can substitute this into the formula to get:
A = (2 + 3t) * (2(2 + 3t) - 4)
Simplifying this expression:
A = (2 + 3t) * (4 + 6t - 4)
A = (2 + 3t) * (6t)
Using the distributive property:
A = 2(6t) + 3t(6t)
A = 12t + 18t^2
Now, we have expressed the area A in terms of t as A = 12t + 18t^2.
To find the rate of change of the area at the instant when t = 2, we need to take the derivative of A with respect to t and evaluate it at t = 2.
Taking the derivative of A = 12t + 18t^2 with respect to t:
dA/dt = 12 + 36t
Substituting t = 2 into the derivative:
dA/dt = 12 + 36(2)
dA/dt = 12 + 72
dA/dt = 84
Therefore, the rate of change of the area at the instant when t = 2 is 84 cm^2/s.