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We are tasked with calculating the length of the cylinder in the given shape, where a cone is attached to the base of a cylinder. The total surface area is provided as \( 335\pi \, \text{cm}^2 \), and the radius \( r \) of both the cylinder and cone is \( 5 \, \text{cm} \), while the slant height \( l \) of the cone is \( 16 \, \text{cm} \).

### Step 1: Surface Area of the Cone
The surface area of the cone is given by its curved surface area:
\[
A_{\text{cone}} = \pi r l
\]
Substituting \( r = 5 \, \text{cm} \) and \( l = 16 \, \text{cm} \):
\[
A_{\text{cone}} = \pi \times 5 \times 16 = 80\pi \, \text{cm}^2
\]

### Step 2: Surface Area of the Cylinder
The surface area of the cylinder is made up of two parts:
1. The side (lateral) surface area of the cylinder, which is:
\[
A_{\text{cylinder}} = 2\pi r h
\]
where \( h \) is the height (or length) of the cylinder.
2. The circular base of the cylinder, which is:
\[
A_{\text{base}} = \pi r^2
\]
Since the base of the cylinder is covered by the cone, we do not need to include the base in the total surface area.

### Step 3: Total Surface Area
The total surface area of the shape is given by the sum of the surface area of the cone and the side surface area of the cylinder:
\[
A_{\text{total}} = A_{\text{cone}} + A_{\text{cylinder}}
\]
We know that:
\[
A_{\text{total}} = 335\pi \, \text{cm}^2
\]
Substitute the expression for \( A_{\text{cone}} \):
\[
335\pi = 80\pi + 2\pi r h
\]
Simplify the equation:
\[
335\pi - 80\pi = 2\pi r h
\]
\[
255\pi = 2\pi \times 5 \times h
\]
Cancel out \( \pi \) from both sides:
\[
255 = 10h
\]
Solve for \( h \):
\[
h = \frac{255}{10} = 25.5 \, \text{cm}
\]

### Final Answer:
The length of the cylinder is \( \mathbf{25.5 \, \text{cm}} \).

Would you like further clarification or more details on this solution?

Here are five related questions to consider:
1. How is the slant height related to the height of the cone?
2. How would the surface area change if the radius of the cylinder were increased?
3. How do you calculate the volume of this composite shape?
4. How would you calculate the total surface area if the base of the cylinder was not covered?
5. How does the lateral surface area of the cylinder depend on its height?

**Tip:** Always remember to separate the contributions from different parts of composite shapes when calculating surface area.

Calculate the length of the cylinder in a shape formed by a cone and cylinder with a total surface area of 335π cm². The radius of both the cone and cylinder is 5 cm, and the slant height of the cone is 16 cm.

Learn how to calculate the length of a cylinder attached to a cone base, given a total surface area of 335π cm². This problem involves calculating surface areas of cones and cylinders, using key formulas like πrl and 2πrh. Suitable for students in Grades 8-10.

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