Bacteria are growing in a circular colony one bacterium thick. the bacteria are growing at a constant rate, making the area of the colony increase at a rate of 12mm^2 per hour. How fast is the radius changing when the radius is 3mm?
A = pi r^2
dA/dt = pi (2r) dr/dt
12 = pi (2*3) dr/dt
dr/dt = 2/pi mm/hr
The length of a telescope has a radius of 1 meter what is the len's circumference use 3.14 for
To determine how fast the radius is changing when the radius is 3mm, we can use the formula for the rate of change of area of a circle with respect to its radius.
The formula for the rate of change of the area of a circle with respect to the radius (A) is:
dA/dt = 2πr * dr/dt
Given that the area of the colony is increasing at a rate of 12mm^2 per hour (dA/dt = 12mm^2/hour), and we want to find the rate of change of the radius (dr/dt) when the radius is 3mm, we can substitute these values into the formula and solve for dr/dt.
dA/dt = 2πr * dr/dt
12 = 2π(3) * dr/dt
12 = 6π * dr/dt
dr/dt = 12 / (6π)
dr/dt = 2 / π
Therefore, when the radius is 3mm, the rate of change of the radius is 2 / π mm per hour.
To find out how fast the radius is changing, we need to use the formula for the rate of change of the area with respect to time. The area of a circular colony can be calculated using the formula:
A = πr^2
Where A is the area and r is the radius.
The problem states that the area of the colony is increasing at a constant rate of 12mm^2 per hour. Let's denote this rate as dA/dt.
We want to find how fast the radius is changing, which is dr/dt. To do this, we need to use the chain rule of calculus. The chain rule states that:
dA/dt = dA/dr * dr/dt
We already know that dA/dt = 12mm^2/hr. So, we need to find dA/dr in order to solve for dr/dt.
Differentiating both sides of the area equation with respect to r, we get:
dA/dr = 2πr
Now we can substitute the given values into the equation:
12mm^2/hr = 2πr * dr/dt
Since we want to find dr/dt when the radius is 3mm, we can substitute r = 3 into the equation:
12mm^2/hr = 2π(3) * dr/dt
Simplifying the equation further:
12mm^2/hr = 6π * dr/dt
Now, rearranging the equation to solve for dr/dt:
dr/dt = (12mm^2/hr) / (6π) = 2mm/hrπ
So, when the radius is 3mm, the rate of change of the radius is approximately 2mm/hrπ.