Final answer:
To find the height and base of the side of the tent, we can use the formula for the area of a triangle, which is 1/2 times base times height. Using this formula, we can set up an equation and solve for the height and base.
Step-by-step explanation:
To find the height and base of the side of the tent, we can use the formula for the area of a triangle, which is 1/2 times base times height. Let's assume the height of the triangle is h.
According to the problem, the base is 6 feet shorter than twice the height, so the base can be represented as 2h - 6. We are given that the area of the triangle is 54 square feet, so we can set up the equation: 1/2 times (2h - 6) times h = 54. Solving this equation will give us the height and base of the triangle.
To solve the equation, we can multiply both sides by 2 to eliminate the fraction: (2h - 6) times h = 108. Expanding the left side of the equation gives us 2h^2 - 6h = 108. Moving all the terms to one side, we get 2h^2 - 6h - 108 = 0.
Now we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula. Let's use factoring in this case. We can factor out a common factor of 2: 2(h^2 - 3h - 54) = 0. Then we can factor the quadratic expression inside the parentheses: 2(h - 9)(h + 6) = 0. Setting each factor equal to zero, we get two possible values for h: h - 9 = 0 or h + 6 = 0. Solving for h in each equation gives us h = 9 or h = -6. Since the height of the triangle cannot be negative, we discard the solution h = -6.
Therefore, the height of the triangle is h = 9 feet. Substituting this value into the expression for the base, we get the base = 2h - 6 = 2(9) - 6 = 12 feet.