The height of a triangle is $8 x^2-6 x+3$ and the base is $(2 x+4)$. Find the area of the triangle using the formula
Area $=\left(\frac{1}{2}\right) b \cdot h$
Set up the equation to solve for the area of the triangle.
$\text { Area }=\left(\frac{1}{2}\right)(2 x+4)(\square \sqrt{x})$
Answer (1)
Substitute the given expressions for the base and height into the area formula: Area = 2 1 ( 2 x + 4 ) ( 8 x 2 − 6 x + 3 ) .
Expand the expression: Area = 2 1 ( 16 x 3 + 20 x 2 − 18 x + 12 ) .
Simplify the expression: Area = 8 x 3 + 10 x 2 − 9 x + 6 .
The area of the triangle is 8 x 3 + 10 x 2 − 9 x + 6 .
Explanation
Problem Setup We are given the height and base of a triangle and asked to find its area. The formula for the area of a triangle is given by Area = 2 1 bh , where b is the base and h is the height. In this case, the height is h = 8 x 2 − 6 x + 3 and the base is b = 2 x + 4 . We need to substitute these expressions into the area formula and simplify.
Substituting and Expanding Substitute the given expressions for the base and height into the area formula: Area = 2 1 ( 2 x + 4 ) ( 8 x 2 − 6 x + 3 ) Now, we expand the expression: Area = 2 1 ( 16 x 3 − 12 x 2 + 6 x + 32 x 2 − 24 x + 12 ) Combine like terms inside the parentheses: Area = 2 1 ( 16 x 3 + 20 x 2 − 18 x + 12 ) Finally, distribute the 2 1 to each term:
Simplifying the Expression Area = 8 x 3 + 10 x 2 − 9 x + 6 So, the area of the triangle is 8 x 3 + 10 x 2 − 9 x + 6 .
Finding the Missing Expression The question asks to set up the equation to solve for the area of the triangle. We have already found the area in terms of x . The equation is: Area = 2 1 ( 2 x + 4 ) ( 8 x 2 − 6 x + 3 ) Comparing this with the given equation: Area = 2 1 ( 2 x + 4 ) ( □ x ) We can see that the missing expression is 8 x 2 − 6 x + 3 . However, the question has a typo, it should be 8 x 2 − 6 x + 3 instead of □ x .
Final Answer Therefore, the area of the triangle is 8 x 3 + 10 x 2 − 9 x + 6 .
Examples
Understanding the area of a triangle is crucial in various fields, such as architecture and engineering. For instance, when designing a roof, calculating the surface area of triangular sections helps determine the amount of material needed. If the height of a triangular section of a roof is given by 8 x 2 − 6 x + 3 meters and the base is 2 x + 4 meters, the area can be calculated using the formula Area = 2 1 bh . This calculation ensures accurate material estimation, preventing waste and reducing costs. By substituting the expressions for base and height, we find the area to be 8 x 3 + 10 x 2 − 9 x + 6 square meters, which aids in efficient planning and resource management.
