A fish tank is 50 cm long, 34 cm wide and 24 cm high. It contains water to depth of 18 cm. Four identical spheres are placed in the tank and are fully submerged. The water level rises by 4.5 cm.
1.What is the volume before the spheres were added?
2. What is the volume of each sphere?
3. What is the radius of the spheres?
Archimedes would have done it this way
original volume
= 50x34x18
=30600
change in volume
= 50x34x4.5
= 7650
---- caused by the two spheres
so each sphere had a volume of 3825 cm^3
V of sphere = (4/3)πr^3
(4/3)πr^3 = 3825
r^3 = 913.151..
r = cuberoot(913.151...) = appr 9.70 cm
1. Well, before the spheres were added, the tank had a length of 50 cm, a width of 34 cm, and a water level of 18 cm. So let's calculate the volume: 50 cm * 34 cm * 18 cm = 30,600 cm³.
2. Now, since we know that the water level rose by 4.5 cm when the spheres were added, we can calculate the volume of each sphere by dividing the total volume increase (4.5 cm) by the number of spheres (4). So, each sphere has a volume of 4.5 cm ÷ 4 = 1.125 cm³.
3. To find the radius of each sphere, we'll use the formula for the volume of a sphere: V = (4/3)πr³. Since we know the volume of each sphere (1.125 cm³), we can rearrange the formula to solve for the radius. Let's do some math magic: 1.125 cm³ = (4/3)πr³. Now we just need to isolate r³: r³ = (1.125 cm³ * 3) / (4π). Finally, we can find the radius by taking the cube root of both sides: r = ∛[(1.125 cm³ * 3) / (4π)].
Now, I'm not too fond of complicated calculations, but I'm sure you can handle it. Good luck finding that radius!
To find the volume before the spheres were added, we need to calculate the volume of the tank without considering the water level.
1. Volume before spheres were added:
The volume of a rectangular prism (fish tank) is calculated by multiplying its length, width, and height.
Volume = Length × Width × Height
Volume = 50 cm × 34 cm × 24 cm
Volume = 40,800 cubic cm
Now, to find the volume of each individual sphere, we need to subtract the volume of the tank without spheres from the total volume after the spheres are added.
2. Volume of each sphere:
The total volume after the spheres are added is the volume of the tank with the increased water level. We can calculate this by finding the difference in water level.
Volume of water after spheres = Volume before + Volume of spheres
Volume of water after spheres = 40,800 cubic cm + 4.5 cm × 50 cm × 34 cm
Volume of water after spheres = 40,800 cubic cm + 7650 cubic cm
Volume of water after spheres = 48,450 cubic cm
Since we have four identical spheres, we can divide the volume of water after spheres by 4 to find the volume of each sphere.
Volume of each sphere = 48,450 cubic cm ÷ 4
Volume of each sphere = 12,112.5 cubic cm
To find the radius of each sphere, we use the formula for the volume of a sphere.
3. Radius of the spheres:
Volume of a sphere = (4/3) × π × radius^3
12,112.5 cubic cm = (4/3) × π × radius^3
To find the radius, we rearrange the formula and solve for it:
radius^3 = (3 × 12,112.5 cubic cm) ÷ (4 × π)
radius^3 = 9,084,375 cubic cm ÷ (4 × π)
radius^3 = 723,504.0071 cubic cm
radius ≈ ∛(723,504.0071)
radius ≈ 92.882 cm (rounded to three decimal places)
Therefore, the radius of each sphere is approximately 92.882 cm.
To find the answers to these questions, we'll need to use some basic geometry and mathematical formulas. Let's start with the first question:
1. What is the volume before the spheres were added?
To find the volume of the tank before the spheres were added, we can multiply the length, width, and height of the tank. The formula for finding the volume of a rectangular prism (like a fish tank) is:
Volume = length * width * height
In this case, the length is 50 cm, the width is 34 cm, and the height is 24 cm. Plugging in these values into the formula, we get:
Volume = 50 cm * 34 cm * 24 cm = 40,800 cm³
Therefore, the volume of the tank before the spheres were added is 40,800 cm³.
2. What is the volume of each sphere?
To find the volume of each sphere, we need to know the change in water level when the spheres were added. In this case, the water level rose by 4.5 cm.
The volume of each sphere can be calculated using the formula:
Volume = (4/3) * π * r³
where r represents the radius of the sphere.
Since the water level rose by 4.5 cm, and the volume of water displaced by the spheres is equal to the volume of the spheres, we can use this information to find the volume of each sphere.
The volume of water displaced is equal to the change in water level multiplied by the base area of the tank. The base area of the tank is given by the length multiplied by the width. So, we have:
Volume of water displaced = change in water level * length * width
Volume of water displaced = 4.5 cm * 50 cm * 34 cm = 7650 cm³
Since there are four spheres and the volume of each sphere is the same, we can divide the total volume of water displaced by 4 to get the volume of each sphere:
Volume of each sphere = Volume of water displaced / 4
Volume of each sphere = 7650 cm³ / 4 = 1912.5 cm³
Therefore, the volume of each sphere is 1912.5 cm³.
3. What is the radius of the spheres?
To find the radius of the spheres, we can rearrange the volume formula and solve for the radius:
Volume = (4/3) * π * r³
Dividing both sides of the equation by (4/3) * π, we get:
Volume / ((4/3) * π) = r³
Taking the cube root of both sides of the equation, we get:
r = ∛(Volume / ((4/3) * π))
Plugging in the volume of each sphere (1912.5 cm³) into the formula, we have:
r = ∛(1912.5 cm³ / ((4/3) * π))
Calculating this, the radius of each sphere is approximately 7.5 cm.
Therefore, the radius of the spheres is approximately 7.5 cm.