Problem 84 - Elementary Algebra

Question

The height of a triangle is \(3 \mathrm{ft}\) more than the length of its base, and its area is \(54 \mathrm{ft}^{2}\). Use a quadratic equation to find the base and height of this triangle. \(\left(A=\frac{1}{2} b h .\right)\)

Step-by-Step Solution

Answer

The base is \( 9 \) feet and the height is \( 12 \) feet.

Step 1 - Define Variables

Let the base of the triangle be \( b \) feet. Then the height is \( b + 3 \) feet.

Step 2 - Write the Area Formula

The area of the triangle is given by \[ A = \frac{1}{2} b h. \]

Step 3 - Substitute Values into the Area Formula

Substitute the values for area \( 54 \) feet\textsuperscript{2}, base \( b \), and height \( b + 3 \) into the formula: \[ 54 = \frac{1}{2} b (b + 3) \]

Step 4 - Simplify the Equation

Multiply both sides by 2 to get rid of the fraction: \[ 108 = b (b + 3) \]

Step 5 - Expand and Rearrange the Equation

Expand the equation and move all terms to one side to form a quadratic equation: \[ b^2 + 3b - 108 = 0 \]

Step 6 - Solve the Quadratic Equation

Solve the quadratic equation using the quadratic formula \[ b = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 1 \), \( b = 3 \), and \( c = -108 \).

Step 7 - Calculate the Discriminant

Calculate the discriminant: \( 3^2 - 4(1)(-108) = 9 + 432 = 441 \)

Step 8 - Find the Roots

Substitute the discriminant back into the quadratic formula to find \( b \): \[ b = \frac{-3 \pm \sqrt{441}}{2} \]. The solutions are \[ b = \frac{-3 + 21}{2} = 9 \] (since base cannot be negative, ignore the negative solution).

Step 9 - Find the Height

Since the height is \( b + 3 \), substitute \( b = 9 \): \[ h = 9 + 3 = 12 \] feet.

Key Concepts

Area of a Triangle

The area of a triangle is a key concept in geometry. To find the area, you use the formula: \[ A = \frac{1}{2} b h \]Here, 'A' stands for area, 'b' is the base of the triangle, and 'h' is the height. This formula tells us that the area of a triangle is half the product of its base and height.
For our problem, the area is given as 54 square feet.
We need to find the values of base and height that make the area equal to 54. Using the formula, we substitute the area along with our base and height values to set up our quadratic equation.

Defining Variables

Defining variables is crucial in solving any math problem. It helps to translate a real-world scenario into a mathematical equation.
For this triangle problem, we choose the base of the triangle to be 'b' feet. According to the problem, the height is '3 feet more than the length of its base'.
Thus, we represent the height as 'b + 3' feet.

• Base, \( b = b \) feet
• Height, \( h = b + 3 \) feet

By clearly defining these variables, we can now substitute them into the area formula and solve for 'b'.

Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation. The standard quadratic equation looks like this:
\[ ax^2 + bx + c = 0 \]
To solve for 'x', we use:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In our triangle problem, after simplifying and reorganizing, we get:
\[ b^2 + 3b - 108 = 0 \]
Here, 'a' is 1, 'b' is 3, and 'c' is -108. We then use the quadratic formula to find the root 'b', which ends up being 9 feet, as we only consider positive values for lengths.

Expanding Equations

Expanding equations involves removing parentheses and simplifying the expression. This step is crucial for converting a simplified equation into a standard quadratic form.
When we substitute the variables into the area formula, we get:
\[ 54 = \frac{1}{2} b (b + 3) \]
By multiplying both sides by 2, we remove the fraction:
\[ 108 = b (b + 3) \]
Next, we expand the right-hand side by distributing 'b':
\[ 108 = b^2 + 3b \]
Finally, we rearrange this to form the quadratic equation:
\[ b^2 + 3b - 108 = 0 \]
Once expanded, we can use the quadratic formula to solve for 'b'.