To find the height of the cone, we can use the formula for the volume of a cone:

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

where:

• $V$ is the volume of the cone,
• $r$ is the radius of the base,
• $h$ is the height of the cone.

We are given:

• $V = 84$,
• $r = 6$.

Substitute these values into the formula:

$84 = \frac{1}{3} \pi (6)^2 h$

Calculate $6^2$:

$84 = \frac{1}{3} \pi \cdot 36 \cdot h$

Now, simplify:

$84 = 12 \pi h$

To isolate $h$, divide both sides by $12 \pi$:

$h = \frac{84}{12 \pi} = \frac{7}{\pi}$

Thus, the height of the cone is:

$h = \frac{7}{\pi} \approx 2.23$

So, the height of the cone is approximately 2.23 units.

Would you like more details or have any questions?

Here are some related questions:

1. What would the height be if the radius were doubled?
2. How would the height change if the volume were doubled?
3. What formula is used to calculate the surface area of a cone?
4. How is the height affected by changes in the cone's radius?
5. What is the effect on volume if both radius and height are doubled?

Tip: Remember that the volume of a cone is directly proportional to both the square of its radius and its height.