A cone shaped container with its vertex at the top and its circular base on the ground is filled to 3/4 of its height with water. What fractional part of the container is filled with water?
consider the unfilled part of the cone.
It will be similar to the original cone but its dimension are 1/4 of the large cone
the volumes of similar shapes are proportional to the cube of their corresponding dimensions
So the volume of the smaller unfilled cone is 1/4^3 or 1/64 of the large cone
so if the height is 3/4 filled with water, that would be 63/64 of the volume of the cone
To find the fractional part of the container that is filled with water, we need to compare the volume of water in the cone to the total volume of 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 a mathematical constant approximately equal to 3.14, r is the radius of the base of the cone, and h is the height of the cone.
In this case, the cone is filled to 3/4 of its height. To calculate the volume of the water in the cone, we first need to determine the height of the water.
Let's assume that the total height of the cone is h, and the height of the water is 3/4 * h.
Now, we can calculate the volume of the water by substituting the given values into the formula:
V_water = (1/3) * π * r^2 * (3/4 * h)
To find the volume of the whole cone, we substitute the total height (h) into the formula:
V_cone = (1/3) * π * r^2 * h
Now, we can calculate the fractional part of the container filled with water by dividing the volume of the water by the volume of the entire cone:
Fraction = V_water / V_cone
By simplifying the expression, we will get the answer.