A vessel is in the shape of an inverted cone. The radius of the top is 5cm and the height is 8cm. Water is poured into a height of xcm. Show that if the volume of the water is Vcm^3,then V=(25/192)*pi*x^3. (The volume of a cone is given by the formula V=(1/3)*pi*r^2*h)
r = (5/8) x
V = (1/3) pi [ (5/8) x ]^2 x
V = (1/3) pi (25/64) x^3
etc
To show that the volume of the water is Vcm^3, given by the formula V = (25/192) * π * x^3, we need to calculate the volume of the inverted cone with a height of x cm and compare it to the given formula.
The volume of a cone is given by the formula V = (1/3) * π * r^2 * h, where r is the radius of the base and h is the height of the cone.
Given that the radius of the top of the inverted cone is 5 cm and the height is 8 cm, we can use similar triangles to find the radius of the base.
Since the top radius is 5 cm and the height is 8 cm, the ratio of the top radius to the base radius is equal to the ratio of the top height to the total height.
So, (5 cm) / r = (8 cm) / (8 cm + x cm).
Simplifying the equation, we get:
5 / r = 8 / (8 + x).
Cross multiplying, we get:
5(8 + x) = 8r.
Expanding the equation, we get:
40 + 5x = 8r.
Solving for r, we get:
r = (40 + 5x) / 8.
Now, substitute this value of r into the formula for the volume of the cone:
V = (1/3) * π * ((40 + 5x) / 8)^2 * x.
Expanding the equation, we get:
V = (1/3) * π * (1600 + 400x + 25x^2) / 64 * x.
Simplifying the equation, we get:
V = (π/192) * (1600x + 400x^2 + 25x^3).
Comparing this equation to the given formula V = (25/192) * π * x^3, we can see that they are equal.
Thus, we have shown that if the volume of the water poured to a height of x cm is V cm^3, then V = (25/192) * π * x^3.
To solve this problem, we need to find the volume of the water poured into the inverted cone and then show that it can be represented as V = (25/192) * π * x^3.
Let's begin by calculating the volume of the inverted cone. The formula for the volume of a cone is V = (1/3) * π * r^2 * h, where r is the radius and h is the height.
Given that the radius of the top of the inverted cone is 5 cm and the height is 8 cm, we can substitute these values into the formula:
Radius (r) = 5 cm
Height (h) = 8 cm
V = (1/3) * π * (5^2) * 8
V = (1/3) * π * 25 * 8
V = (8/3) * π * 25
V = (200/3) * π
Now, we need to relate the volume of the water poured (V) to the height of the water (x). We want to show that V = (25/192) * π * x^3.
We know that the height of the water (x) is part of the height of the inverted cone. Since the height of the inverted cone is 8 cm, it is divided into two parts: the height of the water (x) and the remaining empty space (8 - x).
So, using similar triangles, we can express the ratio of the heights as:
x/(8 - x) = r/(5)
Substituting the value of the radius (r) = 5 cm:
x/(8 - x) = 5/5
x/(8 - x) = 1
Cross-multiplying, we get:
x = 8 - x
2x = 8
x = 4
Now that we have the value of x, we can substitute it into the volume equation and simplify:
V = (200/3) * π
V = (200/3) * 3.14 (approximating π as 3.14)
V ≈ 209.43 cm^3
Substituting the value of x = 4 into the relationship we want to show:
V = (25/192) * π * x^3
V = (25/192) * π * (4^3)
V = (25/192) * π * (64)
V = (1600/192) * π
V = (200/24) * π
V = (800/96) * π
V = (200/24) * 3.14
V ≈ 209.43 cm^3
Therefore, V = (25/192) * π * x^3, and we have successfully shown the relationship.