Final answer:
To find the value of x, set up an equation using the formula for the area of a triangle and solve using factoring. The value of x is 3/2 or 1.5. The actual base of the triangle is 7 feet and the height is 4.5 feet.
Step-by-step explanation:
To find the value of x, we can set up an equation using the formula for the area of a triangle:
A = 1/2 * base * height
Substituting the given values:
27 = 1/2 * (4x + 1) * (3x)
Multiplying both sides of the equation by 2:
54 = (4x + 1) * (3x)
Expanding the equation:
54 = 12x^2 + 3x
Setting the equation equal to zero:
12x^2 + 3x - 54 = 0
Now we can use factoring, completing the square, or the quadratic formula to solve for x. Let's use factoring:
12x^2 + 3x - 54 = (4x - 6)(3x + 9) = 0
Setting each factor equal to zero:
4x - 6 = 0 or 3x + 9 = 0
Solving for x:
4x = 6 or 3x = -9
x = 6/4 or x = -9/3
Therefore, x = 3/2 or x = -3
Since we are looking for a positive value for x, the value of x is 3/2 or 1.5.
To find the actual base and height of the triangle, we can substitute the value of x back into the expressions for the base and height:
Base = 4x + 1 = 4(3/2) + 1 = 6 + 1 = 7 feet
Height = 3x = 3(3/2) = 9/2 = 4.5 feet