SOLUTION: The curve surface area of a cylinder is 1.5 times the total surface area of a cone. If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone i

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Question 1210537: The curve surface area of a cylinder is 1.5 times the total surface area of a cone. If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm. Calculate the height of the cylinder
Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2189) About Me  (Show Source):
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To find the height of the cylinder, we first need to determine the total surface area of the cone and then use the relationship provided to solve for the cylinder's height.
### 1. Find the Dimensions of the Cone
We are given the diameter of the cone is **10 cm**, so the radius () is **5 cm**. The vertical height () is **12 cm**.
To find the Total Surface Area (TSA) of a cone, we first need the **slant height ()**:

### 2. Calculate the Total Surface Area (TSA) of the Cone
The formula for the TSA of a cone is :

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### 3. Calculate the Curved Surface Area (CSA) of the Cylinder
The problem states that the CSA of the cylinder is **1.5 times** the TSA of the cone:

### 4. Solve for the Height of the Cylinder ()
The formula for the Curved Surface Area of a cylinder is . We know the diameter is 10 cm, so the radius () is **5 cm**:

Now, divide both sides by :

**The height of the cylinder is 13.5 cm.**
Would you like me to calculate the volume of either shape to see how they compare?

Answer by ikleyn(53575) About Me  (Show Source):
You can put this solution on YOUR website!
.
The curve surface area of a cylinder is 1.5 times the total surface area of a cone.
If the diameter of the cylinder and cone are equal to 10cm and the vertical height of the cone is 12cm.
Calculate the height of the cylinder
~~~~~~~~~~~~~~~~~~~~~~~~~ 

The curve surface area of this cylinder is  

    A%5Bcylinder%5D = 2%2Api%2Ar%2Ah = 2pi%2A5%2Ah = 10%2Api%2Ah.


The slant height of the cone is  

    L = sqrt%2812%5E2%2B5%5E2%29 = sqrt%28144%2B25%29 = sqrt%28169%29 = 13 cm.


The total surface area of the cone is 

    A%5Bcone%5D = pi%2Ar%2AL + pi%2Ar%5E2 = pi%2A5%2A13 + pi%2A25 = pi%2A%2865%2B25%29 = 90%2Api.


Now write the equation  A%5Bcylinder%5D = 1.5%2AA%5Bcone%5D

    10pi%2Ah = 1.5%2A90%2Api


Simplify it

     h = 1.5*9 =  13.5.


ANSWER.  The height of the cone is 13.5 cm.

Solved.