Let's solve this step-by-step to find the correct equation and the measure of the height of the triangle.
Given:
- Height of the triangle [tex]\( h \)[/tex] is [tex]\( 6c \)[/tex] meters.
- Base of the triangle [tex]\( b \)[/tex] is [tex]\( c-1 \)[/tex] meters.
- Area of the triangle is 18 square meters.
The formula for the area of a triangle is:
[tex]\[ \text{Area} = 0.5 \times \text{base} \times \text{height} \][/tex]
Substitute the given values into the formula:
[tex]\[ 0.5 \times (c-1) \times (6c) = 18 \][/tex]
Let's simplify this equation step-by-step.
1. Multiply the constants and the variable expressions inside the equation:
[tex]\[ 0.5 \times 6c = 3c \][/tex]
So,
[tex]\[ 3c \times (c-1) = 18 \][/tex]
2. Distribute [tex]\( 3c \)[/tex] across [tex]\( (c-1) \)[/tex]:
[tex]\[ 3c^2 - 3c = 18 \][/tex]
3. Move all terms to one side to form a standard quadratic equation:
[tex]\[ 3c^2 - 3c - 18 = 0 \][/tex]
Now, we need to find the positive solution for [tex]\( c \)[/tex] since the length can't be negative. Solving the quadratic equation [tex]\( 3c^2 - 3c - 18 = 0 \)[/tex], we find:
[tex]\[ c = 3 \][/tex]
Next, we find the height of the triangle:
[tex]\[ \text{Height} = 6 \times c \][/tex]
[tex]\[ \text{Height} = 6 \times 3 \][/tex]
[tex]\[ \text{Height} = 18 \][/tex]
Thus, the correct measure of the height of the triangle is 18 meters, and the equation used to solve the problem is:
[tex]\[ 0.5 \times (c-1) \times (6c) = 18 \][/tex]
Therefore, the correct statement is:
[tex]\[ 0.5(c-1)(6 c)=18 ; \text{height } = 18 \text{ meters} \][/tex]
