A glass in the form of a frustum of a cone is represented by the diagram below. The glass contains water to height of 9 cm. The bottom of the glass is circle of radius 2cm while the surface of the water is a circle of radius 6 cm.


(a) Calculate the volume of the water in the glass.
(b) When spherical marble is submerged into the water in the glass, the water level rises by 1 cm. calculate:
(i) The volume of the marble
(ii) The radius of the marble.

using similar triangles, the height of the cone that was cut off to form the glass is 4.5cm. So, the volume of the part cut off is

1/3 pi (2^2)(4.5) = 6pi cm^3

(a) v = 1/3 pi (6^2)(13.5) - 6pi = 156pi
(b)
with marble submerged, volume is increased by (14.5/13.5)^3 = 1.24, or
(i) 37.3pi = 117cm^3
(ii) r = 3cm

Thank you

Could you please explain more on your answer?

To solve this problem, we will use the formulas for the volume of a frustum of a cone and the volume of a sphere.

(a) To calculate the volume of the water in the glass, we need to find the volume of the frustum of the cone formed by the water. The formula for the volume of a frustum of a cone is given by:

V = (1/3) * π * h * (r₁² + r₁ * r₂ + r₂²)

Where V is the volume of the frustum, h is the height of the frustum, and r₁ and r₂ are the radii of the top and bottom circles of the frustum.

In this case, the height of the frustum is the height of the water, which is 9 cm. The radii of the top and bottom circles are 6 cm and 2 cm, respectively.

Plugging these values into the formula, we get:

V = (1/3) * π * 9 * (6² + 6 * 2 + 2²)
V = (1/3) * π * 9 * (36 + 12 + 4)
V = (1/3) * π * 9 * 52
V ≈ 546.5 cm³

So, the volume of the water in the glass is approximately 546.5 cm³.

(b) (i) To calculate the volume of the marble, we need to find the difference in volume before and after the marble is submerged.

The difference in volume is equal to the volume of the marble. Since the water level rises by 1 cm, the volume of water displaced by the marble is equal to the volume of the marble.

The formula for the volume of a sphere is given by:

V = (4/3) * π * r³

Where V is the volume of the sphere and r is its radius.

To find the volume of the marble, we need to use the formula with the radius of the marble, but we don't know the radius yet. Let's call it R.

V₁ = (4/3) * π * R³

This is the volume of the marble before submerging it in water.

(ii) The radius of the marble can be found by rearranging the formula for the volume of a sphere:

V = (4/3) * π * r³

Since we know the volume of the marble (which is equal to the volume of water displaced), we can set up the equation:

V₁ = (4/3) * π * R₁³ (before submerging)
V₂ = (4/3) * π * R₂³ (after submerging)

where V₁ is the volume of the marble before and V₂ is the volume after submerging. R₁ is the radius before and R₂ is the radius after.

We know that the water level rises by 1 cm, so the volume before and after submerging can be expressed as:

V₂ - V₁ = π * (6² + 6 * 2 + 2²)

The value of V₂ - V₁ is the volume of the marble, which we can now substitute in:

(4/3) * π * R₂³ - (4/3) * π * R₁³ = π * (6² + 6 * 2 + 2²)

Simplifying, we get:

(4/3) * π * (R₂³ - R₁³) = π * (52)

Cancelling out the π on both sides, we have:

(R₂³ - R₁³) = 52/ (4/3)

Simplifying, we get:

(3/4) * (R₂³ - R₁³) = 52

Now we can solve for R₂³ - R₁³:

R₂³ - R₁³ = (52 * 4) / 3
R₂³ - R₁³ = 208/3

Lastly, we can take the cube root to find the value of R₂ - R₁:

R₂ - R₁ = (208/3)^(1/3)

So, the radius of the marble can be calculated by subtracting the radius before submerging (2 cm) from R₂:

R = 2 + (208/3)^(1/3)

This is the radius of the marble.