Answer :
To find the height of a triangular window when given certain conditions, let's break it down step-by-step:
1. Identify the Given Data:
- The area of the triangular window is 42 square meters.
- The base of the triangle is 2 meters shorter than twice the height. This can be written as: base = 2 * height - 2.
2. Recall the Formula for the Area of a Triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by the formula:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
3. Set Up the Equation Using the Given Information:
- Substitute the expression for the base into the area formula:
[tex]\[
42 = \frac{1}{2} \times (2h - 2) \times h
\][/tex]
4. Simplify and Arrange the Equation:
- Eliminate the fraction by multiplying the entire equation by 2:
[tex]\[
84 = (2h - 2) \times h
\][/tex]
- Distribute the [tex]\( h \)[/tex] across the terms inside the parenthesis:
[tex]\[
84 = 2h^2 - 2h
\][/tex]
5. Rearrange to Form a Quadratic Equation:
- Bring all terms to one side of the equation to set it to zero:
[tex]\[
2h^2 - 2h - 84 = 0
\][/tex]
- Divide the entire equation by 2 to simplify:
[tex]\[
h^2 - h - 42 = 0
\][/tex]
6. Solve the Quadratic Equation:
- You can solve this quadratic using the quadratic formula, factoring, or other methods, but the height [tex]\( h \)[/tex] that satisfies this equation is:
[tex]\[
h = 7 \text{ meters}
\][/tex]
Therefore, the height of the triangular window is 7 meters.
1. Identify the Given Data:
- The area of the triangular window is 42 square meters.
- The base of the triangle is 2 meters shorter than twice the height. This can be written as: base = 2 * height - 2.
2. Recall the Formula for the Area of a Triangle:
- The area [tex]\( A \)[/tex] of a triangle is given by the formula:
[tex]\[
A = \frac{1}{2} \times \text{base} \times \text{height}
\][/tex]
3. Set Up the Equation Using the Given Information:
- Substitute the expression for the base into the area formula:
[tex]\[
42 = \frac{1}{2} \times (2h - 2) \times h
\][/tex]
4. Simplify and Arrange the Equation:
- Eliminate the fraction by multiplying the entire equation by 2:
[tex]\[
84 = (2h - 2) \times h
\][/tex]
- Distribute the [tex]\( h \)[/tex] across the terms inside the parenthesis:
[tex]\[
84 = 2h^2 - 2h
\][/tex]
5. Rearrange to Form a Quadratic Equation:
- Bring all terms to one side of the equation to set it to zero:
[tex]\[
2h^2 - 2h - 84 = 0
\][/tex]
- Divide the entire equation by 2 to simplify:
[tex]\[
h^2 - h - 42 = 0
\][/tex]
6. Solve the Quadratic Equation:
- You can solve this quadratic using the quadratic formula, factoring, or other methods, but the height [tex]\( h \)[/tex] that satisfies this equation is:
[tex]\[
h = 7 \text{ meters}
\][/tex]
Therefore, the height of the triangular window is 7 meters.
