Beer cans are right circular cylinders. My algebraic ale and logarithmic lager cans, from snide brewery, have dimensional units called brewskis. The have a volume (in cubic brewskis) equal to the solution of the number of grams of radium 226 remaining after 4860 years when the original sample contained 648 pi grams. The height divided by the radius of my beer can is equal to the solution of: log6 W + log6 (w+9)=2. Do not round pi while finding the surface area of my beer can.
I find dumb and cutesy word problems like this very irritating, and will not answer them.
Tell your teacher to give you problems that are meaningful and serve a purpose.
especially since you have to go look up the half-life of 226Ra, which is 1601 years
however, I'm both bored and not proud, so here goes:
v = 648π*(1/2)^(4860/1601) =~ 79π
h/r = w = 3
πr^2*h = 79π
πr^2(3r) = 79π
3r^3 = 79
r = 2.98
a = 2πr(r+h) = 2π*2.98(2.98*4) =~ 71π
I see that the radium decay is suspiciously close to 3 half-lives.
If that was what was wanted, then
v = 81π
r = 3
a = 72π
To find the surface area of your beer can, we need to know the height and radius of the can.
Given that the height divided by the radius of your beer can is equal to the solution of the equation log6 W + log6 (W + 9) = 2, we can solve for W using logarithmic properties.
Here's how to solve it step by step:
Step 1: Combine the logarithms using the product rule:
log6(W(W + 9)) = 2
Step 2: Rewrite the equation in exponential form:
6^2 = W(W + 9)
Step 3: Simplify the equation:
36 = W^2 + 9W
Step 4: Rearrange the equation to bring it to standard form:
W^2 + 9W - 36 = 0
Step 5: Factor the quadratic equation:
(W + 12)(W - 3) = 0
Step 6: Solve for W:
W = -12, W = 3
Since the height divided by the radius of the beer can cannot be negative, we discard W = -12 and only consider W = 3.
Now that we have the value of W, we can proceed to calculate the surface area of the beer can using the formula for the lateral surface area of a cylinder:
Surface Area = 2πr(h + r),
Given that the volume of the beer can is equal to the solution of the number of grams of radium 226 remaining after 4860 years when the original sample contained 648π grams, we need to find the volume of the can first.
The volume V of a cylinder is given by the formula:
V = πr^2h
Let's calculate the volume using the given information:
V = πr^2h = 648π,
Since we are given that the volume is equal to the solution of the number of grams of radium 226 remaining after 4860 years, we need to solve for the number of grams of radium 226 remaining after 4860 years.
The decay of radium 226 follows the formula:
N(t) = N0 * e^(-kt),
where N(t) is the amount remaining at time t, N0 is the initial amount, k is the decay constant, and e is the base of the natural logarithm.
After 4860 years, the remaining amount is given by:
N(4860) = N0 * e^(-kt),
where N0 is the initial amount (648π grams).
To find the decay constant k, we need to know the half-life of radium 226. Once we have the half-life, we can calculate k using the formula:
k = ln(2) / Half-life.
Please provide the half-life of radium 226 so that we can continue with the calculation.