The ratio between the slant height and perpendicular height of a cone is 5:4.The radius of the base of the cone is 6cm.How to find slant height of cone?

let the perpendicular height be 4x

let the slant height be 5x

(4x)^2 + 6^2 = (5x^2
16x^2 + 36 = 25x^2

carry on, of course you would reject the negative value of x after taking the square root at the end.

To find the slant height of a cone, we can use the given ratio between the slant height and the perpendicular height. Here's how to do it:

1. Write down the given information:
- Ratio of slant height to perpendicular height: 5:4
- Radius of the base of the cone: 6 cm

2. Use the formula for the slant height of a cone in terms of the radius and perpendicular height:
slant height (l) = √(radius^2 + height^2)

3. We know the ratio between the slant height and perpendicular height is 5:4, which means l/h = 5/4. Let's substitute the values into the equation:
l/h = 5/4
√(r^2 + h^2) / h = 5/4

4. Substitute the given radius into the equation:
√(6^2 + h^2) / h = 5/4

5. Square both sides of the equation to remove the square root:
(6^2 + h^2) / h^2 = (5/4)^2
(36 + h^2) / h^2 = (25/16)

6. Cross-multiply to get rid of the fractions:
16(36 + h^2) = 25h^2

7. Expand and simplify the equation:
576 + 16h^2 = 25h^2
25h^2 - 16h^2 = 576
9h^2 = 576

8. Divide both sides of the equation by 9:
h^2 = 576 / 9
h^2 = 64

9. Take the square root of both sides to find the perpendicular height:
h = √64
h = 8 cm

10. Substitute the value of h back into the original equation, to find the slant height:
l/h = 5/4
l/8 = 5/4
l = 8 * (5/4)
l = 10 cm

Therefore, the slant height of the cone is 10 cm.