Determine the slant height of a right pyramid whose lateral area is 90 ft^2 and whose base is a regular hexagon with a 3 ft side.
To determine the slant height of the pyramid, we can use the lateral area and the properties of a regular hexagon.
Given that the lateral area is 90 ft², we know that the sum of the areas of all six triangular faces is 90 ft².
Since the base of the pyramid is a regular hexagon, it can be divided into six congruent equilateral triangles. The side length of each of these triangles is 3 ft.
To calculate the area of an equilateral triangle, we can use the formula:
Area = (side length² * √3) / 4
Substituting the given side length as 3 ft, we find:
Area = (3² * √3) / 4
= (9 * √3) / 4
= (9√3) / 4
Since there are six triangles, the sum of their areas is:
90 ft² = 6 * [(9√3) / 4]
90 ft² = (54√3) / 4
To simplify the expression, we can multiply both the numerator and denominator by 4:
90 ft² * 4 = 54√3
360 ft² = 54√3
To isolate the square root, divide both sides of the equation by 54:
(360 ft²) / 54 = √3
(20 ft²/3) = √3
Finally, square both sides of the equation to get rid of the square root:
[(20 ft²/3)²] = (√3)²
400 ft⁴/9 = 3
Now, take the square root of both sides to isolate the slant height:
√(400 ft⁴/9) = √3
20 ft²/3 = √3
Therefore, the slant height of the pyramid is equal to (20 ft²/3) or approximately 6.113 ft.