The interior of a typical mesung cup is a right circular cylinder of radius 6 cm. The volume of water we put in the cup is therefore a function of the level h to which the cup is filled. How closely do we have to measure h to measure out one liter of water with an error of no more than 1% (i.e. 10 cm3)?
The cylinder cross section area is
pi*r^2 = 113.1 cm^2
The height of a column of liquid containing a volume of 1000 cm^3 is
h = 1000/113.1 = 8.84 cm
For a volume measurment accuracy of 1%, the height measurement accuracy must also be 1%, or in this, case 0.09 cm
To determine how closely we need to measure the level, h, to measure out one liter of water with an error of no more than 1%, we need to calculate the volume of one liter and find the corresponding range of values for h.
1 liter is equal to 1000 cm^3.
The volume, V, of a right circular cylinder is given by the formula:
V = πr^2h,
Where r is the radius of the cylinder and h is the height or level to which the cylinder is filled.
In our case, the radius, r, is given as 6 cm.
So, the formula becomes:
V = π(6^2)h
V = 36πh
To measure one liter of water with an error of no more than 1%, we can calculate the upper and lower limits of the volume:
Upper Limit = 1000 + 10 cm^3 = 1010 cm^3
Lower Limit = 1000 - 10 cm^3 = 990 cm^3
Now, let's substitute these values back into our formula to find the upper and lower limits for h:
1010 = 36πh
h_upper = 1010 / (36π)
990 = 36πh
h_lower = 990 / (36π)
Calculate the values of h_upper and h_lower:
h_upper ≈ 8.97 cm
h_lower ≈ 8.77 cm
Therefore, we need to measure the level, h, with an error of no more than approximately 0.2 cm (h_upper - h_lower) to measure out one liter of water with an error of no more than 1% (10 cm^3).