Sure! Let's break down the problem step-by-step to find which equation models the given situation.
1. Assign variables:
- Let the base of the triangle be [tex]\( x \)[/tex] feet.
- The height of the triangle is given as 1 more than 6 times its base. Therefore, the height is [tex]\( 6x + 1 \)[/tex] feet.
2. Use the area formula for 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]
- Substitute the given values into this formula. The area is 13 square feet:
[tex]\[
13 = \frac{1}{2} \times x \times (6x + 1)
\][/tex]
3. Simplify the equation:
- To eliminate the fraction, multiply both sides by 2:
[tex]\[
26 = x \times (6x + 1)
\][/tex]
- Distribute [tex]\( x \)[/tex] on the right side:
[tex]\[
26 = 6x^2 + x
\][/tex]
4. Rearrange the equation:
- Move all terms to one side of the equation to set it to zero:
[tex]\[
6x^2 + x - 26 = 0
\][/tex]
Hence, the correct equation that models the situation is:
[tex]\[
D. \quad 6x^2 + x - 26 = 0
\][/tex]
Therefore, the correct answer is:
D. [tex]\(36 x^2 + 6 x - 26=0\)[/tex]
