A chemist pours 200 grams of a liquid (density = 2.5 gm/mL) into a cube beaker with 5 cm edges and a graduated cylinder with a base radius of 5 cm and a height of 5 cm.
What is fraction of the liquid that needs to be put in the graduated cylinder such that the liquid height is the same in the beaker
and graduated cylinder?
The area of the graduated cylinder (g) floor = pi*5*5 = 25*pi.
Th area of the beaker (b) floor is 5*5 = 25.
(Area)g/(Area)b = 25*pi/25 = pi
Since the area of the graduated cylinder is larger than that of the beaker, the volume poured into the graduated cylinder must be pi times volume of that poured into the beaker.
Let x = volume poured into the beaker, then 80-x = volume poured into the graduated cylinder, and
pi*x = 80-x
Solve for x, then x/80 = fraction.
I get x = about 19 and 80-x = about 61.
Check and solve for h for grad cyl = pi*r^2*h and
solve for h for beaker from V= 25*h; i.e.,
h = Vg/pi*25 = about 0.8
h = Vb/25 = about 0.8
You can do it more accurately. Check my work. I'm not a math man. Perhaps Reiny or Mathmate will check it for us.
Thanks! I understand what you are doing. Just one question: how did you get 80?
Also, Is the height of the liquid in the cube beaker less than or greater than 1 cm? This was a follow up question. Since, the answer is 0.8. This would be less than, correct?
The actual values I came up with were 19.316 mL and 60.684 mL (although that is far too many significant figures).
height = volume/pi*25 for graduated cylinder
height = 60.694/25*pi = 0.773 cm height.
height = volume/25 for cubic beaker = 19.316/25 =0.773 cm height.
I rounded those in my first post to 19/25 = 0.76 which is about 0.8 and
61/pi*25 = 0.78 whichis about 0.8.
To answer this question, we need to compare the volume of the liquid poured into the beaker with the volume of the graduated cylinder.
First, let's calculate the volume of the liquid poured into the beaker. We know that the density of the liquid is 2.5 g/mL. Since the mass is given as 200 grams, we can use the formula:
Volume = Mass / Density
Volume = 200 g / 2.5 g/mL = 80 mL
Next, we need to find the volume of the graduated cylinder. The volume of a cylinder can be calculated using the formula:
Volume = π * radius^2 * height
The base radius of the cylinder is given as 5 cm, and the height is also 5 cm. So, plugging these values into the formula:
Volume = π * 5^2 * 5 = 125π cm^3 (approximately 392.7 mL)
Now, to find the fraction of the liquid that needs to be put in the graduated cylinder, we need to compare the volumes:
Fraction = Volume of Graduated Cylinder / Volume of Liquid Poured
Fraction = (392.7 mL) / (80 mL) ≈ 4.91
Therefore, approximately 4.91 times the volume of the liquid poured into the beaker needs to be put in the graduated cylinder to have the same liquid height in both containers.