Let's start by using the formula for the area of a circle:
A = πr^2
where A is the area and r is the radius.
We are given that a circular ripple is created in a lake and it travels outward at a speed of 70 cm/s. This means that the radius of the circle is increasing at a rate of 70 cm/s.
(a) After 1 second, the radius of the circle is:
r = 70 cm/s × 1 s = 70 cm
Now we can use the formula for the area of a circle to find the rate at which the area within the circle is increasing:
A = πr^2
A = π(70 cm)^2
A ≈ 15,400 cm^2
To find the rate at which the area within the circle is increasing, we need to take the derivative of the area with respect to time:
dA/dt = 2πr(dr/dt)
where dA/dt is the rate at which the area is increasing, and dr/dt is the rate at which the radius is increasing (which we know is 70 cm/s).
Plugging in the values we have:
dA/dt = 2π(70 cm)(70 cm/s)
dA/dt ≈ 9,800 cm^2/s
So after 1 second, the area within the circle is increasing at a rate of approximately 9,800 cm^2/s.