The vase company designs a new vase that is shaped like a cylinder on bottom and a cone on top. The catalog states that the width is 12 cm and the total height is 42 cm. What would the height of the cylinder part have to be in order for the total volume to be 1224pi cm cubed?
volume=PI*(diameter/2)^2 *h + pi(diamter/2)^2 (1/3)(42-h)
solve for h.
To find the height of the cylinder part in order for the total volume to be 1224π cm³, we first need to calculate the volume of the vase.
The volume of a cylinder is given by the formula:
V_cylinder = π * r² * h,
where r is the radius of the base of the cylinder and h is the height of the cylinder.
The volume of a cone is given by the formula:
V_cone = (1/3) * π * r² * h_cone,
where h_cone is the height of the cone.
Since the vase is shaped like a cylinder on the bottom and a cone on top, the total volume of the vase can be found by summing up the volumes of the cylinder and cone:
V_total = V_cylinder + V_cone.
Given that the width (diameter) of the vase is 12 cm, the radius of the cylinder's base is 12/2 = 6 cm.
We also know that the total height of the vase is 42 cm. Let's assume that the height of the cylinder part is h_cylinder.
Therefore, the height of the cone part, h_cone, would be the difference between the total height and the height of the cylinder:
h_cone = 42 - h_cylinder.
Now, let's substitute the values into the volume formulas:
V_cylinder = π * (6)² * h_cylinder
V_cone = (1/3) * π * (6)² * (42 - h_cylinder)
The total volume of the vase is set to be 1224π cm³:
V_total = 1224π
Now, we can combine all the equations and solve for h_cylinder:
V_total = V_cylinder + V_cone
1224π = π * (6)² * h_cylinder + (1/3) * π * (6)² * (42 - h_cylinder)
Simplifying the equation:
1224 = 36 * h_cylinder + (1/3) * 36 * (42 - h_cylinder)
1224 = 36h_cylinder + 12(42 - h_cylinder)
1224 = 36h_cylinder + 504 - 12h_cylinder
1224 = 24h_cylinder + 504
24h_cylinder = 720
h_cylinder = 720 / 24
h_cylinder = 30
Therefore, the height of the cylinder part would need to be 30 cm in order for the total volume to be 1224π cm³.
To find the height of the cylinder part of the vase, we first need to determine the volume of the vase and then use that information to solve for the height of the cylinder.
The volume, V, of the vase can be calculated by summing the volumes of the cylinder and cone parts.
The volume of a cylinder is given by the formula V_cylinder = πr^2h_cylinder, where r is the radius of the circular base and h_cylinder is the height of the cylinder.
The volume of a cone is given by the formula V_cone = (1/3)πr^2h_cone, where h_cone is the height of the cone.
Since the vase is described as having a cylinder on the bottom and a cone on top, with the same base width of 12 cm, the radius of both the cylinder and cone is half of the width, which is 12/2 = 6 cm.
Given that the total volume of the vase is 1224π cm^3, we can set up the equation:
V_cylinder + V_cone = 1224π
Now, let's substitute the formulas for V_cylinder and V_cone:
π(6^2)h_cylinder + (1/3)π(6^2)h_cone = 1224π
Simplifying the equation by canceling out π and squaring 6, we get:
36h_cylinder + 12h_cone = 1224
Since the total height of the vase is given as 42 cm, we can write the equation:
h_cylinder + h_cone = 42
Now, we have a system of two equations with two unknowns:
36h_cylinder + 12h_cone = 1224
h_cylinder + h_cone = 42
To solve this system of equations, we can use substitution or elimination methods. Since we need to find the height of the cylinder, we'll use the substitution method.
Rearranging the second equation, we have:
h_cone = 42 - h_cylinder
Substituting this value for h_cone in the first equation, we get:
36h_cylinder + 12(42 - h_cylinder) = 1224
Simplifying the equation:
36h_cylinder + 504 - 12h_cylinder = 1224
Combining like terms:
24h_cylinder + 504 = 1224
Moving 504 to the other side of the equation:
24h_cylinder = 720
Dividing both sides by 24:
h_cylinder = 720 / 24
Simplifying:
h_cylinder = 30
Therefore, the height of the cylinder part of the vase must be 30 cm in order for the total volume to be 1224π cm^3.