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To solve this problem, we need to use the formulas for the surface area and volume of a cone. The information provided includes the following:

- Diameter of the base = \( 10a \) cm, so the radius \( r \) is half of that:  
  \( r = \frac{10a}{2} = 5a \) cm.
  
- Height \( h \) of the cone = \( 12a \) cm.
  
- Surface area \( A \) of the cone = \( 360\pi \) cm².
  
- Volume of the cone \( V = k\pi \) cm³, where \( k \) is an integer.

### Step 1: Surface Area of the Cone
The total surface area of a cone is given by:

\[
A = \pi r(r + l)
\]
where:
- \( r \) is the radius,
- \( l \) is the slant height, calculated by \( l = \sqrt{r^2 + h^2} \).

Now, substitute \( r = 5a \) and \( h = 12a \):

\[
l = \sqrt{(5a)^2 + (12a)^2} = \sqrt{25a^2 + 144a^2} = \sqrt{169a^2} = 13a.
\]

So, the surface area becomes:

\[
A = \pi r (r + l) = \pi \times 5a \times (5a + 13a) = \pi \times 5a \times 18a = 90a^2 \pi.
\]

We know the total surface area is \( 360\pi \), so we can set up the equation:

\[
90a^2 \pi = 360\pi
\]

Dividing both sides by \( \pi \):

\[
90a^2 = 360
\]

Solve for \( a^2 \):

\[
a^2 = \frac{360}{90} = 4
\]

Thus, \( a = 2 \) (since \( a^2 = 4 \)).

### Step 2: Volume of the Cone
The volume of a cone is given by:

\[
V = \frac{1}{3}\pi r^2 h
\]

Substituting \( r = 5a \) and \( h = 12a \), with \( a = 2 \):

\[
V = \frac{1}{3} \pi (5 \times 2)^2 (12 \times 2) = \frac{1}{3} \pi (10)^2 (24) = \frac{1}{3} \pi (100) (24) = \frac{1}{3} \pi (2400) = 800\pi \, \text{cm}^3.
\]

Thus, \( k = 800 \).

### Final Answer:
The value of \( k \) is \( \boxed{800} \).

Would you like any additional details or have any questions?

Here are 5 related questions for further practice:
1. How would the problem change if the height were doubled while keeping the diameter the same?
2. How does the slant height affect the surface area of a cone?
3. If the surface area were given as \( 180\pi \) cm², what would be the new value of \( a \)?
4. What is the ratio of the volume to surface area of the cone?
5. If the diameter were halved, what would be the new volume of the cone?

**Tip:** For any cone, the slant height is crucial for calculating the surface area because it includes both the base and the lateral curved surface.

The diameter of the base of the cone is 10a cm. The height of the cone is 12a cm. The total surface area of the cone is 360π cm². The volume of the cone is kπ cm³, where k is an integer. Find the value of k.

This problem involves finding the volume and surface area of a cone with a base diameter of 10a cm and a height of 12a cm. With the total surface area given as 360π cm², we determine the value of k for the cone's volume. Suitable for students in Grades 9-12.

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