Answer :
To determine which equation could be used to find the length of the tomato patch given the area of the vegetable garden is 170 square feet, let's evaluate each provided equation step by step.
1. Equation 1: \(0 = 3x^2 - 160\)
This equation implies that when solved for \(x\), we are considering a scenario where the area of the garden would be a form that solves \(3x^2 = 160\). However, if the actual given area is 170 square feet, this equation does not directly relate to the area of 170 square feet.
2. Equation 2: \(0 = 3x^2 + 17x - 160\)
To verify if this equation can be used, we need to check if it can somehow represent the 170 square feet area. This equation stands a better chance since it has a term that could potentially balance out the 170 to find an appropriate \(x\). It integrates terms that account for both quadratic and linear contributions, thus fitting more realistic scenarios for certain geometric configurations where \(x\) may represent lengths contributing to an area closer to 170 square feet.
3. Equation 3: \(0 = 3x^2 + 2x + 180\)
This equation suggests solving for \(x\) in a context where \(3x^2 + 2x = -180\). This would not be synonymous with a positive area of 170, hence it could not represent the situation where the area is 170 square feet.
4. Equation 4: \(0 = 3x^2 + 10x + 180\)
Similarly, this equation would solve for \(x\) in a scenario where \(3x^2 + 10x = -180\), clearly not aligning with our known area of 170 square feet.
Thus, by evaluating these options and matching the given area of 170 square feet, we deduce that the appropriate and valid equation which can be employed to find the length of the tomato patch is:
[tex]\[ \boxed{0 = 3x^2 + 17x - 160} \][/tex]
1. Equation 1: \(0 = 3x^2 - 160\)
This equation implies that when solved for \(x\), we are considering a scenario where the area of the garden would be a form that solves \(3x^2 = 160\). However, if the actual given area is 170 square feet, this equation does not directly relate to the area of 170 square feet.
2. Equation 2: \(0 = 3x^2 + 17x - 160\)
To verify if this equation can be used, we need to check if it can somehow represent the 170 square feet area. This equation stands a better chance since it has a term that could potentially balance out the 170 to find an appropriate \(x\). It integrates terms that account for both quadratic and linear contributions, thus fitting more realistic scenarios for certain geometric configurations where \(x\) may represent lengths contributing to an area closer to 170 square feet.
3. Equation 3: \(0 = 3x^2 + 2x + 180\)
This equation suggests solving for \(x\) in a context where \(3x^2 + 2x = -180\). This would not be synonymous with a positive area of 170, hence it could not represent the situation where the area is 170 square feet.
4. Equation 4: \(0 = 3x^2 + 10x + 180\)
Similarly, this equation would solve for \(x\) in a scenario where \(3x^2 + 10x = -180\), clearly not aligning with our known area of 170 square feet.
Thus, by evaluating these options and matching the given area of 170 square feet, we deduce that the appropriate and valid equation which can be employed to find the length of the tomato patch is:
[tex]\[ \boxed{0 = 3x^2 + 17x - 160} \][/tex]
