Final answer:
To solve for the base and height of the triangle, use the given information of the area and the height being two less than six times the base. Set up and solve an equation to find the base and height using the quadratic formula. Round the answers to the nearest tenth.
Step-by-step explanation:
To solve for the base and height of the triangle, we can use the given information that the area is 70 square inches and the height is two less than six times the base. Let's denote the base as 'x' and the height as '6x - 2'. We can then use the formula for the area of a triangle (A = 1/2 * base * height) and substitute the given values:
70 = 1/2 * x * (6x - 2)
To solve for 'x', we can multiply both sides of the equation by 2 to eliminate the fraction:
140 = x * (6x - 2)
Expanding the equation:
140 = 6x^2 - 2x
Rearranging the equation to make it quadratic:
6x^2 - 2x - 140 = 0
Now, we can solve for 'x' using factoring, square root principle, or quadratic formula. Let's use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, 'a' is 6, 'b' is -2, and 'c' is -140:
x = (-(-2) ± sqrt((-2)^2 - 4*6*(-140))) / (2*6)
Simplifying further:
x = (2 ± sqrt(4 + 3360))/12
x = (2 ± sqrt(3364))/12
x = (2 ± 58.037)/12
Dividing each x value by 12:
x ≈ 5.836 or x ≈ -0.119
Since a triangle cannot have a negative length, we discard the negative value. Therefore, the approximate value of the base 'x' is 5.836 inches.
To find the height, we substitute the value of 'x' back into the equation for height: height = 6x - 2
height = 6(5.836) - 2 ≈ 32.016 inches