Question 1210537
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?