Final answer:
To find the base and height of the triangle, we can use the formula for the area of a triangle and solve for the variables. The height of the triangle is 20.5 inches and the base is 48 inches.
Step-by-step explanation:
To solve this problem, we can use the formula for the area of a triangle: Area = 1/2 x base x height. Let's assign variables for the base and height. Let's say the height is h, and the base is 2h + 7 (since the base is seven more than twice the height). We can substitute these values into the area formula: 277.5 = 1/2 x (2h + 7) x h. Now we can solve the equation.
First, distribute the 1/2 to (2h + 7): 277.5 = (h + 7/2) x h. Next, simplify: 277.5 = h^2 + 7/2h. Putting the equation in standard form: h^2 + 7/2h - 277.5 = 0.
We can solve this quadratic equation using factoring, completing the square, or quadratic formula. The factors of -277.5 are -13.5 and 20.5. So we have two possible solutions for h: h = -13.5 or h = 20.5. Since height cannot be negative, we discard the negative solution. Therefore, the height of the triangle is 20.5 inches.
To find the base, we substitute the value of h into the expression for the base: base = 2h + 7 = 2(20.5) + 7 = 41 + 7 = 48 inches. Therefore, the base of the triangle is 48 inches and the height is 20.5 inches.