Final answer:
To find the length of the base of the pyramid, we can use the area formula for a triangle. By substituting the area and slant height into the formula, we can rearrange the equation to solve for the base. The length of the base of the pyramid is 8 meters.
Step-by-step explanation:
To find the length of the base of the pyramid, we can use the formula for the area of a triangle. The formula is Area = 1/2 × base × height. In this case, the area of the triangle is given as 64m² and the slant height is given as 8 meters. Since we don't know the height of the triangle, which is the height of the pyramid, we cannot directly calculate the length of the base.
However, we can use the formula for the slant height of a pyramid to find the height of the triangle. The formula is Slant Height² = Height² + (1/4) × Base². Substituting the given values, we get 8² = Height² + (1/4) × Base².
Simplifying the equation, we get 64 = Height² + (1/4) × Base². Since we know the area of the triangle is 64m², we can substitute Height × Base ÷ 2 for the area in the equation. This gives us 64 = Height² + (1/4) × Base² = Height × Base ÷ 2 + (1/4) × Base². To solve for the length of the base, we rearrange the equation as 0 = Height × Base ÷ 2 + (1/4) × Base² - 64.
We can now solve this quadratic equation for the unknown length of the base using the quadratic formula. After substituting the values into the formula and simplifying, we get two possible solutions: Base = -16 and Base = 8. Only the positive solution, Base = 8, is valid since length cannot be negative. Therefore, the length of the base of the pyramid is 8 meters.