Question

a skyscraper has a triangular window with an area of 42 square meters. the window’s base is 2 meters shorter than twice the height. which equation can you use to find h, the height of the window in meters? \\(\frac{1}{2}(2h - 2)h = 42\\) \\((2h - 2)h = 42\\) \\(\frac{1}{2}(2h)h = 42\\) \\(\frac{1}{2}(h - 2)h = 42\\) \\(\frac{1}{2}(2h + 2)h = 42\\) \\(\frac{1}{2}(2h + 2)(h - 2) = 42\\) now, use the equation you picked to find h. \\(h = \\) meters

Explanation:

Part 1: Choosing the Correct Equation

Step 1: Recall the area formula for a triangle

The area \( A \) of a triangle is given by \( A = \frac{1}{2} \times \text{base} \times \text{height} \).

Step 2: Define the base in terms of height

Let \( h \) be the height of the window. The base is 2 meters shorter than twice the height, so the base \( b = 2h - 2 \).

Step 3: Substitute into the area formula

Substitute \( b = 2h - 2 \) and \( A = 42 \) into the area formula:
\[
\frac{1}{2}(2h - 2)h = 42
\]

Part 2: Solving for \( h \)

Step 1: Simplify the equation

Start with \( \frac{1}{2}(2h - 2)h = 42 \). Multiply both sides by 2 to eliminate the fraction:
\[
(2h - 2)h = 84
\]
Expand the left side:
\[
2h^2 - 2h = 84
\]
Subtract 84 from both sides to set the equation to zero:
\[
2h^2 - 2h - 84 = 0
\]
Divide all terms by 2 to simplify:
\[
h^2 - h - 42 = 0
\]

Step 2: Factor the quadratic equation

Factor \( h^2 - h - 42 \). We need two numbers that multiply to -42 and add to -1. These numbers are -7 and 6:
\[
(h - 7)(h + 6) = 0
\]

Step 3: Solve for \( h \)

Set each factor equal to zero:
\[
h - 7 = 0 \quad \text{or} \quad h + 6 = 0
\]
Solving these gives \( h = 7 \) or \( h = -6 \). Since height cannot be negative, we discard \( h = -6 \).

Part 1 Answer:

The correct equation is \(\boldsymbol{\frac{1}{2}(2h - 2)h = 42}\)

Part 2 Answer:

\( h = \boldsymbol{7} \) meters

Answer:

Step 1: Simplify the equation

Start with \( \frac{1}{2}(2h - 2)h = 42 \). Multiply both sides by 2 to eliminate the fraction:
\[
(2h - 2)h = 84
\]
Expand the left side:
\[
2h^2 - 2h = 84
\]
Subtract 84 from both sides to set the equation to zero:
\[
2h^2 - 2h - 84 = 0
\]
Divide all terms by 2 to simplify:
\[
h^2 - h - 42 = 0
\]

Step 2: Factor the quadratic equation

Factor \( h^2 - h - 42 \). We need two numbers that multiply to -42 and add to -1. These numbers are -7 and 6:
\[
(h - 7)(h + 6) = 0
\]

Step 3: Solve for \( h \)

Set each factor equal to zero:
\[
h - 7 = 0 \quad \text{or} \quad h + 6 = 0
\]
Solving these gives \( h = 7 \) or \( h = -6 \). Since height cannot be negative, we discard \( h = -6 \).

Part 1 Answer:

The correct equation is \(\boldsymbol{\frac{1}{2}(2h - 2)h = 42}\)

Part 2 Answer:

\( h = \boldsymbol{7} \) meters