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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 [tex]h[/tex], the height of the window in meters?

A. [tex]\frac{1}{2}(2h)h = 42[/tex]
B. [tex]\frac{1}{2}(2h + 2)(h - 2) = 42[/tex]
C. [tex](2h - 2)h = 42[/tex]
D. [tex]\frac{1}{2}(2h + 2)h = 42[/tex]
E. [tex]\frac{1}{2}(h - 2)h = 42[/tex]
F. [tex]\frac{1}{2}(2h - 2)h = 42[/tex]

Now, use the equation you picked to find [tex]h[/tex].

[tex]h = \square[/tex] meters

Answer :

We start by noting that for a triangle, the area is given by

[tex]$$
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}.
$$[/tex]

According to the problem, the area of the window is 42 square meters and the base is 2 meters shorter than twice the height. If we let the height be [tex]$h$[/tex], then the base can be expressed as

[tex]$$
\text{base} = 2h - 2.
$$[/tex]

Thus, we can set up the equation using the area formula:

[tex]$$
\frac{1}{2}(2h - 2)h = 42.
$$[/tex]

This is the correct equation to find [tex]$h$[/tex].

To solve for [tex]$h$[/tex], we first eliminate the fraction by multiplying both sides by 2:

[tex]$$
(2h - 2)h = 84.
$$[/tex]

Expanding the left side gives:

[tex]$$
2h^2 - 2h = 84.
$$[/tex]

Subtract 84 from both sides to form a quadratic equation:

[tex]$$
2h^2 - 2h - 84 = 0.
$$[/tex]

Dividing every term by 2 simplifies the equation to:

[tex]$$
h^2 - h - 42 = 0.
$$[/tex]

Next, we calculate the discriminant of the quadratic equation. Recall that for the quadratic equation [tex]$ah^2 + bh + c = 0$[/tex], the discriminant is given by

[tex]$$
\Delta = b^2 - 4ac.
$$[/tex]

Here, [tex]$a = 1$[/tex], [tex]$b = -1$[/tex], and [tex]$c = -42$[/tex]. Therefore,

[tex]$$
\Delta = (-1)^2 - 4(1)(-42) = 1 + 168 = 169.
$$[/tex]

Since [tex]$\sqrt{169} = 13$[/tex], the solutions for [tex]$h$[/tex] are given by

[tex]$$
h = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{1 \pm 13}{2}.
$$[/tex]

This yields the two solutions:

[tex]$$
h = \frac{14}{2} = 7 \quad \text{or} \quad h = \frac{-12}{2} = -6.
$$[/tex]

Because [tex]$h$[/tex] represents a height, it must be a positive number. Therefore, we discard [tex]$h = -6$[/tex] and accept

[tex]$$
h = 7 \quad \text{meters}.
$$[/tex]

In summary, the equation used is

[tex]$$
\frac{1}{2}(2h-2)h = 42,
$$[/tex]

and the height of the window is

[tex]$$
h = 7 \text{ meters}.
$$[/tex]

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