A cylinder of radius 14cm contains water.A metal solid cone of base radius 7cm and height 18cm is submerged into water. Find the change in height of the water level in the cylinder
the water displaced has volume
v = π/3 r^2 h = π/3 * 7^2 * 18 = 2401π cm^3
so now you can find h using
196πh = 2401π
Well, let's dive into this problem and see what's going on. We have a cylinder with some water and we're introducing a solid cone into it. In order to find the change in height of the water level, we need to consider the volume of the cone and how much water it displaces.
Now, the volume of a cone is given by the formula V = (1/3)πr²h, where r is the base radius and h is the height. Using this formula, we can calculate the volume of the cone.
V_cone = (1/3)π(7cm)²(18cm)
Now, since the cone is submerged in water, it will displace some of the water. The volume of water displaced by the cone is equal to the volume of the cone itself. So, the change in height of the water level in the cylinder will be equal to the height of the cone, which is 18cm.
Therefore, the change in height of the water level in the cylinder is 18cm.
I hope this explanation didn't leave you all wet!
To find the change in height of the water level in the cylinder, we need to determine the volume of the cone, which will be equal to the volume of water displaced by the cone.
The volume of a cone can be calculated using the formula:
V = (1/3) * π * r^2 * h
where V is the volume, π is the mathematical constant pi (approximately 3.14159), r is the base radius of the cone, and h is the height of the cone.
Given that the base radius of the cone is 7 cm and the height of the cone is 18 cm, we can substitute these values into the formula to find the volume:
V = (1/3) * 3.14159 * 7^2 * 18
= (1/3) * 3.14159 * 49 * 18
= 3.14159 * 49 * 6
≈ 919.875 cm^3
Since the volume of the cone is equal to the volume of water displaced, the change in height of the water level in the cylinder can be found by dividing the volume of the cone by the base area of the cylinder.
The base area of the cylinder can be calculated using the formula:
A = π * r^2
where A is the base area and r is the radius of the cylinder.
Given that the radius of the cylinder is 14 cm, we can substitute this value into the formula to find the base area:
A = 3.14159 * 14^2
= 3.14159 * 196
≈ 615.752 cm^2
Finally, we can find the change in height of the water level in the cylinder by dividing the volume of the cone by the base area of the cylinder:
Change in height = V / A
= 919.875 cm^3 / 615.752 cm^2
≈ 1.494 cm
Therefore, the change in height of the water level in the cylinder is approximately 1.494 cm.
To find the change in height of the water level in the cylinder, we need to determine the volume of the cone and the volume of the water displaced by the cone.
Step 1: Find the volume of the cone:
The volume of a cone can be calculated using the formula: V = (1/3)πr^2h, where V is the volume, π is approximately 3.14, r is the radius of the base, and h is the height of the cone.
In this case, the base radius (r) is 7 cm, and the height (h) is 18 cm. Plug these values into the formula to calculate the volume of the cone.
V_cone = (1/3) * π * 7^2 * 18
V_cone ≈ 1/3 * 3.14 * 49 * 18
V_cone ≈ 2588.08 cm³ (rounded to two decimal places)
Step 2: Find the volume of the water displaced by the cone:
The volume of a cylinder can be calculated using the formula: V = πr^2h, where V is the volume, π is approximately 3.14, r is the radius of the cylinder, and h is the height of the water displaced.
In this case, the cylinder has the same base radius (14 cm) as the cone. We need to calculate the change in height of the water level, so let's represent this change as Δh.
V_water = π * 14^2 * Δh
V_water ≈ 3.14 * 196 * Δh
V_water ≈ 615.44Δh (rounded to two decimal places)
Step 3: Equate the volumes of the cone and the water displaced:
Since the volume of the cone that is submerged into water is equal to the volume of the water displaced, we can set up the equation: V_cone = V_water.
2588.08 = 615.44Δh
Step 4: Solve for Δh:
Divide both sides of the equation by 615.44 to isolate Δh.
Δh ≈ 2588.08 ÷ 615.44
Δh ≈ 4.2 cm (rounded to one decimal place)
Therefore, the change in height of the water level in the cylinder is approximately 4.2 cm.