Bacteria are growing in a circular colony one bacterium thick. The bacteria are growing at a constant rate, thus making the area of the colony increase at a constant rate of 12 mm sq. per hour. Find an equation expressing the rate of change of the radius as a function of the radius, r, in millimeters, of the colony. Plot dr/dt as a function of r. How fast is r hanging when it equals 3 mm? Describe the way dr/dt changes with the radius of the circle.
We just started learning related rates problems like the one above but I'm totally stumped on this one.
area=PI8r^2
darea/dt=PI*2 r dr/dt
you are given darea/dt=12mm^2/hr
so dr/dt= 12/(2r) mm/hr
I'm sorry, I'm not following what you did. I get that we know dx/dt = 12mm^2/hr and we want to find dr/dt but I don't know how you linked them together as a function of r. Maybe show me more steps and how you got the derivative dr/dt. Thanks
wow, you are lost.
Start with the relationship
A=PI*r^2
take the derivative with respect to time:
dA/dt=PI*2*r*dr/dt
you are given dA/dt. Solve for dr/dt
Why are you stuck with using x as a variable? The world is more generous than that.
12mm^2/hr = r2PI dr/dt
So 12/2PIr mm^2/hr = dr/dt
which is the same as dr/dt = 6/PIr mm^2/hr
Why'd you originally say dr/dt= 12/(2r) mm/hr
I must have dropped a PI.
now all u got to do is plug in the 3 for r and u will get the answer
To solve this problem, let's start by considering the relationship between the area of the colony and its radius.
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius. We know that the area of the colony is increasing at a constant rate, so we can express this as dA/dt = 12 mm^2/hr.
We want to find the rate of change of the radius with respect to time, dr/dt, in terms of the radius, r. To do this, we can use implicit differentiation.
Take the derivative of both sides of the equation A = πr^2 with respect to time:
dA/dt = d(πr^2)/dt
The derivative of A with respect to time is dA/dt, and the derivative of πr^2 with respect to time is 2πr(dr/dt). This is because we are treating r as a function of time, so we need to use the chain rule.
Now we have:
12 mm^2/hr = 2πr(dr/dt)
Divide both sides of the equation by 2πr to solve for dr/dt:
(dr/dt) = 12 mm^2/hr / (2πr)
Simplifying further, we find:
(dr/dt) = 6 / (πr)
Now we have our equation expressing the rate of change of the radius as a function of the radius itself.
To plot dr/dt as a function of r, you can create a graph with r on the x-axis and dr/dt on the y-axis. For each value of r, calculate dr/dt using the equation (dr/dt) = 6 / (πr). This will give you a curve showing how the rate of change of the radius varies with the radius of the colony.
To find how fast r is changing when it equals 3 mm, substitute r = 3 mm into the equation (dr/dt) = 6 / (πr). Solve for dr/dt to get the rate of change of the radius at that specific point.
Regarding the way dr/dt changes with the radius of the circle, we can observe from the equation (dr/dt) = 6 / (πr) that as r increases, dr/dt decreases. This means that initially, when the radius is small, the rate of change of the radius is high, but as the colony grows, the rate of change decreases. This makes intuitive sense because as the colony grows, there are more bacteria contributing to the increase in area, so the individual contribution of each bacterium decreases.