To determine which equation could be used to find the length of the tomato patch given that the area of the vegetable garden is 170 square feet, let's analyze each of the provided equations.
1. [tex]\( 0 = 3x^2 + 2x + 180 \)[/tex]
2. [tex]\( 0 = 3x^2 + 17x - 160 \)[/tex]
3. [tex]\( 0 = 3x^2 + 10x + 180 \)[/tex]
4. [tex]\( 0 = 3x^2 - 160 \)[/tex]
Since the goal is to find out which equation could be used to represent a situation involving an area of 170 square feet, we need to determine which equation fits this context. Given the area is provided, an equation should align with the characteristic balance for such a practical domain problem.
By closely evaluating the mentioned equations:
- The first equation [tex]\(0=3x^2+2x+180\)[/tex] does not correspond correctly with any suitable reduction involving the area of 170 square feet.
- The third equation [tex]\(0=3x^2+10x+180\)[/tex] similarly isn't aligning appropriately when setting logical parameters that involve 170.
- The fourth equation [tex]\(0=3x^2-160\)[/tex] lacks the necessary construction that would involve or encompass variables tying directly into 170 effectively.
Upon deeper assessment and algebraic exploration, the second equation [tex]\(0 = 3x^2 + 17x - 160\)[/tex] provides a correct, balanced equation that effectively offers a plausible functional form fitting the area specified, adjusting variables matched perfectly to the vegetable garden area in question.
Thus, the correct equation that can be used to find the length of the tomato patch, given the area of 170 square feet, is:
[tex]\[ 0 = 3x^2 + 17x - 160 \][/tex]
