Final answer:
To find the slant height and radius of the cone, Pythagorean theorem and the volume formula for a cone are used. After expressing the radius and slant height in terms of a common multiplier and simplifying the volume equation, it is found that the radius is approximately 5 meters and the slant height is approximately 12 meters.
Step-by-step explanation:
To find the slant height and radius of a cone where the radius (r) and slant height (l) have a ratio of 5:12, and the volume (V) is 314 cubic meters, we use the volume formula for a cone:
V = 1/3 π r² h
However, to proceed, we need the height (h), which can be found by using the Pythagorean theorem in the right-angled triangle formed by the radius, height, and slant height. Since we are given the ratio of r to l, we can express the radius as r = 5x and the slant height as l = 12x, where x is a common multiplier. Consequently, the height (h) can be expressed as:
h = √(l² - r²) = √((12x)² - (5x)²) = √(144x² - 25x²) = √(119x²) = 10.9x (approximately)
Substituting r = 5x into the volume formula, we get:
314 = 1/3 π (5x)² (10.9x) = 1/3 π 25x³ 10.9
Simplifying and solving for x gives us the common multiplier, which can then be used to calculate the radius and the slant height. By solving, we can find that the radius is approximately 5 meters and the slant height is approximately 12 meters.