HOw many ounces of Pepsi remain? Assume that a can holds exactly 12 ounces and that there is no variation in filling or construction of the can. Also assume a can is a perfect cyldiner with no rounded edges, that it is exactly 4.75 inches high, and 2 inches in diameter. Assume the center most part of the opening is exactly .875 inches from the edge of the can. Use these numbers only, they may not converst correctly.
Okay, that clears that up. But I'm not sure about that last part of your statement:
This tells me that the CENTER of the opening is at 0.875 inches, but we don't know the size of the opening. The highest level for the fluid will be at the bottom of the opening, not at the center of it.
-Dan
So basically you want to find the volume of a, uh, right prism, whose length is 4.75 inches and whose cross-section is that of a cirlce that is 2 inches in diameter but that is cut 0.875 inches from one edge.
The Pepsi can is lying on its side horizontally, the opening, 0.875 inches deep from the top of the can, is on the vertical diameter. What remains is the pepsi inside the can, whose level is 0.875 inches from the top of the can.
If full, the can holds exactly 12 ounces of pepsi.
Since the volume of a right cylinder is equal to the area of the circular base times the height, and the volume of the said "right prism" is also area of its cross-section times its length, and since the height or length here is the same 4.75 inches, then the volume of the right prism in ounces is a variation of, or is proportional to, the 12 ounces according to the proportions of the two mentioned areas.
[Do you wonder now why numbers are better to use than words in Math calculations?]
Okay, let's use numbers.
When the circle is "complete":
The area, A, of the circle is pi(2/2)^2 = pi(1) = pi sq.in.
So, the volume, V, of the can = A(4.75) = pi(4.75) = 4.75pi cu.in.
It is given that 4.75pi cu.in. of pepsi is 12 ounces exactly.
When the circle is (2 -0.875) = 1.125 inches high only:
Area = A, minus the area of the secant of the circle that is 0.875 inch high.
The secant of the circle here is the empty space above the level of the remaining pepsi inside the can.
The area of the small secant is (the area of the small sector) minus (the area of the isosceles triangle whose base is the chord of the secant). [Words!]
The area of the small sector is (1/2)*(included angle in radians)*(radius)^2.
The included angle is 2*arccos(0.125/1) = 2.89 radians
So, area of small sector = (1/2)(2.89)(1^2) = 1.445 sq.in.
Using the formula, Area = (1/2)ab*sinC, the area of the isosceles triangle is
= (1/2)*(radius)*(radius)*sin(2.89rad)
= (1/2)(1)(1)(0.249)
= 0.1245 sq.in.
So, the area of the small secant = 1.445 -0.1245 = 1.3205 sq.in.
And so, the area of the cut circle is
= pi(1^2) -1.3205
= 3.1416 -1.3205
= 1.8211 sq.in.
Now, area of full circle = pi(1^2) = 3.1416 sq.in.
So, by proportion,
(12oz)/3.1416 = (x)/1.8211
x = (12oz)(1.8211/3.1416)
x = 6.956 ounces.
Or, x = 7 ounces if rounded off.
Therefore, about 7 ounces of pepsi still remains in the can. ----answer.