I feel really dumb because I cant figuer out this simple algebra I problem...
84. The diameter of a copper (Cu) atom is roughly 1.3 x 10^-12 m. How mnay times can you divide devenly a piece of 10-cm copper wire until it is reduced to two separate copper atoms? (Assume there are appropriate tools fo rthis procedure and that copper atoms are lined up in a stragiht line, in contact with each other. Round off your answer to an integer.)
The back of the book says 36 times which I don't see how...
your are cutting the wire in half, so you are solving
1.3x10^-12 = 1x10^-2 (1/2)^n , where n is the number of times.
1.3x10^-11 = (.5)^n
log[1.3x10^-11] = log(.5^n)
n = log[1.3x10^-11] / log .5
n = 36.16
or
n = 36
my first line has a typo, should have been
1.3x10^-12 = 1x10^-1 (1/2)^n , where n is the number of times.
I agree that 36 times seems fishy to me; however, the answer is correct. The secret word may be "evenly".
How many atoms do we have lined up in 0.1 meter?
1.3 x 10^-12 m/atom x # atoms = 0.1m
Solve for # atoms and I get something like 7.69 x 10^10 but you need to verify that.
Now the first slice cuts it in half (and you now have 3.8 x 10^10 pieces in the half you will cut next.
The second slice leaves 1.9 x 10^10 atoms in your hand.
The third slice leaves 9.6 x 10^9 in your hand.
Continue cutting until you get to 1 piece in your hand.
If I do that 36 times, I'm left with 1.11 pieces in my hand. I don't know what we do with the 0.11 piece of an atom (perhaps those end up as electrons, protons, and neutrons). :-)
Reiny's answer is more esoteric than mine. It's more sophisticated because he used math to solve the problem and I just iterated it. It's easier to punch my divide button 36 times than it is to burn my brain trying to deduce the math.
To solve this problem, we need to figure out how many times we can divide the length of the copper wire by the diameter of a copper atom until we reach a length of just two atoms.
First, let's convert the length of the copper wire from centimeters to meters. We have 10 centimeters, which is equal to 0.1 meters.
Now, let's see how many times we can divide 0.1 meters by the diameter of a copper atom, which is 1.3 x 10^-12 meters.
To calculate the number of divisions, we can use the following equation:
Number of divisions = (Length of copper wire) / (Diameter of copper atom)
First, let's calculate the length of the copper wire in terms of atomic diameters:
Length of copper wire in atomic diameters = (0.1 meters) / (1.3 x 10^-12 meters)
To divide a length by a small number, you can multiply the numerator and denominator by a large power of 10 to simplify the division. In this case, we can multiply both sides of the equation by 10^12 to make the calculation easier:
Length of copper wire in atomic diameters ≈ (0.1 meters) * (10^12 / 1.3)
Now, let's calculate the approximate number of divisions:
Number of divisions ≈ (Length of copper wire in atomic diameters) ≈ (0.1 meters) * (10^12 / 1.3)
Using a calculator, we find that the result is approximately 7.6923076923076923076923076923077 x 10^12 divisions.
Since we need to round off the answer to an integer, the closest whole number is 7.
Therefore, according to the calculations, the copper wire can be divided approximately 7 times, not 36 times as mentioned in the book. It's possible that there is an error in the provided answer.