A rectangular tank with a base measuring 50cm by 24cm contained 3.6 liters of water. When 74 identical marbles were dropped into the tank, the water level rose to a certain height. If 26 marbles were removed from the tank, the new water level become 9cm. Find the volume of each marble.
Firstly, we need to find the original height of the water level before the marbles were dropped in. We can do this by dividing the volume of water (3.6 liters = 3600 mL) by the base area of the tank (50cm x 24cm = 1200 cm^2):
original height = 3600 mL ÷ 1200 cm^2 = 3 cm
Now, when the marbles are dropped in, they displace an amount of water equal to their total volume. Let's assume each marble has a volume of V cm^3. The total volume of the marbles is then:
total marble volume = 74V cm^3
This volume of water is added to the tank, causing the level to rise to a new height. Let's call this new height H cm. The total volume of water and marbles in the tank is now:
total volume = (50cm x 24cm x H cm) + 74V cm^3
But we know that this total volume is equal to the original volume plus the volume of the marbles:
total volume = 3600 mL + 74V cm^3
We can convert the original volume to cm^3 by multiplying by 1 mL / 1 cm^3, so:
total volume = 3600 cm^3 + 74V cm^3
Setting these two expressions for total volume equal to each other, we get:
(50cm x 24cm x H cm) + 74V cm^3 = 3600 cm^3 + 74V cm^3
Simplifying and solving for H, we get:
H = (3600 cm^3 + 74V cm^3) ÷ (50cm x 24cm) = 3 cm + V ÷ 8.11
Now, we can use the second piece of information to solve for V. When 26 marbles are removed, the total volume of water and marbles is reduced to what it was originally (3600 cm^3), and the water level drops from H to 9 cm. Using the same method as before, we can find the original height of the water level:
original height = 3600 mL ÷ 1200 cm^2 = 3 cm
So the change in height is:
change in height = 9 cm - 3 cm = 6 cm
The total volume removed from the tank is equal to the volume of the 26 marbles that were taken out, or:
total removed volume = 26V cm^3
So the volume of water that was originally in the tank is:
original water volume = 3600 cm^3 - 26V cm^3
This volume of water is contained in a rectangular shape with base area (50cm x 24cm) and height 6 cm. The volume of this shape is:
volume = (50cm x 24cm x 6 cm) = 7200 cm^3
Setting this equal to the original water volume, we get:
7200 cm^3 = 3600 cm^3 - 26V cm^3
Solving for V, we get:
V = -138.46 cm^3
This negative volume doesn't make sense, so we must have made an error somewhere. Looking back at the equations, we see that we set up:
(50cm x 24cm x H cm) + 74V cm^3 = 3600 cm^3 + 74V cm^3
But we forgot to subtract the original volume of water from the left side:
(50cm x 24cm x H cm) + 74V cm^3 - 3600 cm^3 = 74V cm^3
Simplifying and solving for H, we get:
H = (74V cm^3) ÷ (50cm x 24cm) + 3 cm = V ÷ 8.11 + 3 cm
Now, when we set up the equation using the second piece of information, we get:
7200 cm^3 = (50cm x 24cm x 6 cm) + 74V cm^3 - 3600 cm^3
Simplifying and solving for V, we get:
V = 30.46 cm^3
Therefore, the volume of each marble is 30.46 cm^3.
First, note that 3.6 liters is equivalent to 3600 cubic centimeters. When 26 marbles were removed from the tank, the volume of the water and the remaining 48 marbles is 50 x 24 x 9 cubic centimeters.
Next, let V be the volume of each marble. The volume of 74 marbles is 74V, so when they are dropped into the tank, the total volume is 3600+74V.
When 26 marbles are removed from the tank, the volume of the water and the remaining 48 marbles is 50 x 24 x 9. To find the volume of 48 marbles, subtract the volume of the water, 3600, from this volume: 50 x 24 x 9 - 3600 = 48V.
Now solve for V:
50 x 24 x 9 - 3600 = 48V
12000 - 3600 = 48V
8400 = 48V
V = 175 cubic centimeters
So, each marble has a volume of 175 cubic centimeters.
original tank :
50 x 24 x h = 3600
h = 3
So 74 marbles were added, then 26 were removed , so in effect
48 marbles are in the tank when the height is 9 cm
volume with the 48 marbles = 50x24x9 = 10800
The change was caused by the 48 marbles
change in volume = 10800-3600 cm^3 = 7200 cm^2
let the volume of 1 marble be V
48V = 7200
V = 7200/48 = 150 cm^3
Now look at the bot's two attempts, the first is a very long litany of gibberish,
at least it realized it made an error by getting a negative answer.
It then tried a second time, and it actually ended up with the
correct equation of
50 x 24 x 9 - 3600 = 48V , which is the same as mine,
BUT, then the bot failed to solve this simple equation correctly,
getting 175 instead of the correct 150 cm^3
You are correct, the volume of each marble is indeed 150 cm^3. Thank you for pointing out the errors in the bot's attempts.
To find the volume of each marble, we need to determine the change in water level caused by the marbles.
1. Start by finding the initial volume of water in the tank. We know that the tank contains 3.6 liters of water. Since 1 liter is equal to 1000 cubic centimeters (cm³), the initial volume of water in the tank is 3.6 x 1000 = 3600 cm³.
2. Find the change in water level caused by the 74 marbles.
- First, determine the volume of the tank. The tank has a base measuring 50cm by 24cm, so the area of the base is 50 x 24 = 1200 cm².
- Multiply the area of the base by the change in water level to find the change in volume: 1200 cm² x height change = volume change.
- Since the water level rose when the marbles were added, the volume change is equal to the volume of the marbles. Let's call the volume of each marble V.
- So, 74V = volume change.
3. Find the new volume of water after 26 marbles are removed.
- We know that when 26 marbles were removed, the new water level became 9 cm.
- Using the same formula as before, the volume change caused by removing the marbles is 1200 cm² x 26 marbles x V = 9 x 1200 = 10800 cm³.
- Therefore, 26V = 10800 cm³.
4. Now we can solve the equations to find the volume of each marble.
- Since we have two equations: 74V = volume change and 26V = 10800 cm³, we can solve them simultaneously.
- Divide the second equation by 26 to get V = 10800 / 26 = 415.38 cm³.
So, the volume of each marble is approximately 415.38 cm³.