Applications of Quadratic Equations

We will use the formula for the area of a triangle to solve the next example.

AREA OF A TRIANGLE

For a triangle with base \(b\) and height \(h\), the area, \(A\), is given by the formula \(A=\frac{1}{2} b h\).

Recall that, when we solve geometry applications, it is helpful to draw the figure.

EXAMPLE 10.39

An architect is designing the entryway of a restaurant. She wants to put a triangular window above the doorway. Due to energy restrictions, the window can have an area of 120 square feet and the architect wants the width to be 4 feet more than twice the height. Find the height and width of the window.

Step 1. Read the problem. Draw a picture.
Step 2. Identify what we are looking for.	We are looking for the height and width.
Step 3. Name what we are looking for.	Let \(h\) = the height of the triangle. Let \(2h+4\) = the width of the triangle
Step 4. Translate.	We know the area. Write the formula for the area of a triangle.
Step 5. Solve the equation. Substitute in the values.
Distribute.
This is a quadratic equation, rewrite it in standard form.
Solve the equation using the Quadratic Formula. Identify the \(a\), \(b\), \(c\).
Write the quadratic equation.
Then substitute in the values of \(a\), \(b\), \(c\).
Simplify.
Simplify the radical.
Rewrite to show two solutions.
Simplify.
Since \(h\) is the height of a window, a value of \(h = -12\) does not make sense.
	The height of the triangle: \(h = 10\) The width of the triangle: \(\begin{gathered} 2 h+4 \\ 2 \cdot 10+4 \\ 24 \end{gathered}\)
Step 6. Check the answer. Does a triangle with a height 10 and width 24 have area 120? Yes.
Step 7. Answer the question.	The height of the triangular window is 10 feet and the width is 24 feet.

Notice that the solutions were integers. That tells us that we could have solved the equation by factoring.

When we wrote the equation in standard form, \(h^{2}+2 h-120=0\), we could have factored it. If we did, we would have solved the equation \((h+12)(h-10)=0\).

TRY IT 10.77

Find the dimensions of a triangle whose width is four more than six times its height and has an area of 208 square inches.

TRY IT 10.78

If a triangle that has an area of 110 square feet has a height that is two feet less than twice the width, what are its dimensions?