Final answer:
The expression for the height of the triangle is derived by simplifying the given area expression and solving for height, resulting in (3x + 7)(x + 15)(5x - 8)/2. The perimeter cannot be determined with the given information. If the height is reduced by a scale factor of 2, the new area expression is x(3x + 7)(x + 15)(5x - 8)/(2 × 2 × 2).
Step-by-step explanation:
Simplifying the Height Expression
To find the expression for the height of the triangle, we'll use the formula for the area of a triangle, which is A = 1/2 × base × height. The given area expression is x(3x + 7)(x + 15)(5x - 8)/(5 + 1).
First, we need to simplify the fraction by dividing the entire expression by 6, which is the denominator of the area formula. Then, since we only have the product of x, (3x + 7), (x + 15), and (5x - 8), that would be twice the area, because the typical expression for the area of a triangle does not include the coefficient of 1/2. So, we divide again by 2 to get the area in terms of base times height. If the base is one of the factors, say x, then we can solve for the height by dividing the area by x, giving us a simplified expression for the height: (3x + 7)(x + 15)(5x - 8)/2.
Calculating the Perimeter
To calculate the perimeter, we need to know the lengths of all sides of the triangle. However, with the information given, we cannot calculate the perimeter because we need individual side lengths, not just the product of terms representing the area of the triangle.
Area with Reduced Height
If the height is reduced by a scale factor of 2, the new height will be half of the original height. Therefore, the new area can be found by multiplying the base by the new height (which is the original height divided by 2) and then dividing by 2 again for the area of a triangle. Thus, the new expression for the area of the triangle will be:
x(3x + 7)(x + 15)(5x - 8)/(2 × 2 × 2).