Final answer:
The radius of the cone is 4.7625 cm, and the slant height of the cone is 7.9375 cm, calculated using the volume formula and the Pythagorean theorem considering the ratio of the radius to the height given.
Step-by-step explanation:
The question is asking us to find the radius and slant height of a cone given that the volume is 301.44 cm³ and the ratio of the radius to height is 3:4. The formula for finding the volume of a cone is V = ⅓πr²h.
Let's denote the radius as 3x and the height as 4x based on the provided ratio. Substituting into the volume formula we get:
301.44 = ⅓π(3x)²(4x).
Simplifying the equation, we find that x³ equals to 4. Therefore, x equals to √{4}, which is 1.5875 cm. Thus, the radius (3x) is 4.7625 cm.
Now, to find the slant height (l), we can use the Pythagorean theorem, since the slant height is the hypotenuse of the right-angled triangle formed by the radius, height, and slant height of the cone. So, l = √{r² + h²} = √{(3x)² + (4x)²} = √{9x² + 16x²} = √{25x²} = 5x. Hence, the slant height is 7.9375 cm.