The area of a blot of ink is growing such that after t seconds its area is given by A=(3t square +7) cm square. Calculate the rate of increase of area at t=5 second
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To find the rate of increase of the area at t = 5 seconds, we need to calculate the derivative of the area function A(t) with respect to time.
The given area function is A = 3t^2 + 7.
To find dA/dt (the derivative of A with respect to t), we differentiate the function A(t) with respect to t:
dA/dt = d/dt (3t^2 + 7)
The derivative of 3t^2 with respect to t is 6t (using the power rule of differentiation). The derivative of a constant (7 in this case) is zero. Therefore, we can simplify the equation:
dA/dt = 6t + 0
Simplifying further, we have:
dA/dt = 6t
Now, substitute t = 5 into the equation dA/dt = 6t to find the rate of increase of the area at t = 5 seconds:
dA/dt at t = 5 seconds = 6 * 5 = 30 cm^2/s
Therefore, the rate of increase of the area at t = 5 seconds is 30 cm^2/s.