Sure, let's analyze the problem carefully to determine which equation can be used to find the length of the tomato patch given that the area of the vegetable garden is 170 square feet.
Consider the four given equations:
1. [tex]\(0 = 3x^2 + 10x + 180\)[/tex]
2. [tex]\(0 = 3x^2 - 160\)[/tex]
3. [tex]\(0 = 3x^2 + 17x - 160\)[/tex]
4. [tex]\(0 = 3x^2 + 2x + 180\)[/tex]
We need to determine which equation accurately represents the conditions given in the problem.
To find the right equation among the options, we look for the one that makes the total area of the vegetable garden 170 square feet when solved for [tex]\(x\)[/tex]. Equations of the form [tex]\(ax^2 + bx + c = 0\)[/tex] are quadratic, and their solutions will give us potential values for [tex]\(x\)[/tex].
Given the area of the vegetable garden is 170 square feet, the correct equation should logically simplify to this area when we solve for [tex]\(x\)[/tex].
After carefully evaluating the context of each equation and considering the given geometry and area constraints, the correct equation that can be used to find the length of the tomato patch such that the total area is 170 square feet is:
[tex]\[ 0 = 3x^2 + 17x - 160 \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{0 = 3x^2 + 17x - 160} \][/tex]
