I need to find the radius of a cylinder with only the following info:
Cylinder height=125um
There is 1 C atom/0.01848 nm^2
There are 5.75x10^10 C atoms on the surface of cylinder
Use quadratic formula to solve
To find the radius of the cylinder, we need to use the information given and the formula for the surface area of a cylinder. The surface area of a cylinder is given by the formula:
Surface Area = 2πrh + 2πr^2
Where r is the radius of the cylinder and h is the height.
First, let's convert the height of the cylinder from micrometers (μm) to nanometers (nm) because the atom density unit is in nm^2. We know that 1 μm = 1000 nm, so the height of the cylinder is 125 μm * 1000 nm/μm, which equals 125,000 nm.
The given information tells us that there are 1 C atom for every 0.01848 nm^2, and there are 5.75x10^10 C atoms on the surface of the cylinder. To find the total surface area of the cylinder in nm^2, we divide the number of C atoms by the atom density:
Surface Area = 5.75x10^10 C atoms / 0.01848 nm^2
Now we have the equation for the surface area of the cylinder:
Surface Area = 2πrh + 2πr^2
Substituting the known values into the equation:
5.75x10^10 / 0.01848 = 2πr * 125,000 + 2πr^2
Simplifying the equation, we get a quadratic equation in terms of r:
πr^2 + 125,000πr - (5.75x10^10 / 0.01848) = 0
To solve this quadratic equation for r, you can use the quadratic formula:
r = (-b ± √(b^2 - 4ac)) / 2a
In this equation, a is the coefficient of r^2, b is the coefficient of r, and c is the constant term. In our case, a = π, b = 125,000π, and c = -(5.75x10^10 / 0.01848).
Plug in these values to the quadratic formula, evaluate the expression, and you will get two possible values for the radius of the cylinder.