A cylinder container of radius 15 cm has some water in it. When a solid submerged into the water, the water level rises 1.2 cm.
(a) Find, the volume of the water displace by the solid leaving your answer in terms of π
(b) If the solid is a circular cone of the height 9 cm, calculate the radius of the cone
v = 225π*1.2
1/3 πr^2 = 9
(a) To find the volume of the water displaced by the solid, we need to calculate the volume of the submerged portion of the solid, which is equal to the volume of water that the cylinder initially had. Let's call this volume V.
The formula for the volume of a cylinder is given by V = πr^2h, where r is the radius and h is the height.
In this case, the radius of the cylinder is 15 cm and the height is the increase in water level, which is 1.2 cm. Therefore, the volume of the water initially in the cylinder is:
V = π(15 cm)^2(1.2 cm)
V = π(225 cm^2)(1.2 cm)
V = 270π cm^3
So, the volume of the water displaced by the solid is 270π cm^3.
(b) Now, let's consider the solid as a circular cone. We are given the height of the cone, which is 9 cm.
The formula for the volume of a cone is given by V = (1/3)πr^2h, where r is the radius of the cone and h is the height.
Since we know the height of the cone and want to find the radius, we can rearrange the formula as follows:
V = (1/3)πr^2h
3V = πr^2h
r^2 = (3V)/(πh)
r = sqrt((3V)/(πh))
Plugging in the values, the radius of the cone is:
r = sqrt((3*270π cm^3)/(π*9 cm))
r = sqrt(810 cm^3/9 cm)
r = sqrt(90 cm^2)
r = 9 cm
Therefore, the radius of the cone is 9 cm.
To find the volume of water displaced by the solid, we need to calculate the difference in volumes before and after placing the solid in the water.
(a) The volume of water displaced by the solid can be found using the formula:
Volume of water displaced = π * r^2 * h
Where:
- π is the mathematical constant pi (approximately 3.14159)
- r is the radius of the cylinder (15 cm)
- h is the rise in water level (1.2 cm)
Substituting the values, we have:
Volume of water displaced = π * (15 cm)^2 * 1.2 cm
Volume of water displaced = π * 225 cm^2 * 1.2 cm
Volume of water displaced = 270π cm^3
Therefore, the volume of the water displaced by the solid is 270π cm^3.
(b) To calculate the radius of the cone, we can use the concept of similar triangles. The cone shape resembles a scaled-down version of the cylinder, with a height of 9 cm compared to the cylinder's height.
If we consider the small triangle formed within the cylinder by connecting the center of the base circle, the water level, and the height of the cone, it is similar to the larger triangle formed by connecting the center of the base circle, the original water level, and the height of the cylinder.
We can set up a proportion between the two triangles to find the relationship between the radius of the cone and the radius of the cylinder:
(height of cone) / (height of cylinder) = (radius of cone) / (radius of cylinder)
Substituting the given values:
9 cm / (9 cm + 1.2 cm) = (radius of cone) / 15 cm
Simplifying the equation:
9 cm / 10.2 cm = (radius of cone) / 15 cm
Cross-multiplying:
9 cm * 15 cm = 10.2 cm * (radius of cone)
Solving for the radius of the cone:
(radius of cone) = (9 cm * 15 cm) / 10.2 cm
(radius of cone) = 13.33 cm (rounded to two decimal places)
Therefore, the radius of the cone is approximately 13.33 cm.