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.
Given that
A = 3t²+ 7
Here we can find out the value of A easily if we differentiate it ,
On doing differentiation we obtain
area = dx/dt
= d ( 3t² + 7 ) / dt = 6t
we are given that t = 5
second
Now putting t = 5
A = 6 × 5
A = 30 cm²
Hope this helps you ☺
A = 3t^2 + 7
dA/dt = 6t
now plug in t=5
To calculate the rate of increase of the area at t=5 seconds, we need to find the derivative of the area function A(t) with respect to time (t) and then plug in t=5.
First, let's find the derivative of A(t):
A'(t) = dA/dt
Given that A(t) = 3t^2 + 7, we can differentiate A(t) with respect to t using the power rule for derivatives:
A'(t) = d/dt(3t^2 + 7)
A'(t) = d/dt(3t^2) + d/dt(7)
Using the power rule, we have:
A'(t) = 6t + 0
A'(t) = 6t
Now that we have the derivative function, we can substitute t=5 to find the rate of increase of the area at t=5 seconds:
A'(5) = 6 * 5
A'(5) = 30
Therefore, the rate of increase of the area at t=5 seconds is 30 cm^2/second.