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Ethan wants to build a porch with a length that is 6 less than three times its width and an area that is at least 200 square feet. Which inequality can be solved to find possible widths for Ethan’s deck?
3w² – 6w ≥ 200
3w² – 6w – 200 ≤ 0
w(3w – 6) ≤ 200
w² ≥ 200
To solve Ethan’s problem, we first establish the relationship between the length (L) and width (w) of the porch. Ethan wants the length to be 6 less than three times the width, which can be expressed as:
L = 3w – 6.
The area (A) of the porch can be calculated using the formula:
A = L * w.
Substituting the expression for L, we get:
A = (3w – 6) * w = 3w² – 6w.
Since we know that the area needs to be at least 200 square feet, we can set up the inequality:
3w² – 6w ≥ 200.
To bring it to a standard form, we rearrange it to:
3w² – 6w – 200 ≥ 0.
Thus, the correct inequality that can be solved to find possible widths for Ethan’s deck is:
3w² – 6w – 200 ≥ 0.
However, this can sometimes be rephrased to solve for when the quadratic expression is less than or equal to 0, resulting in the option that fits best from your provided choices:
3w² – 6w – 200 ≤ 0.
This means we can find the widths that will satisfy the area requirement.
So the answer is 3w² – 6w – 200 ≤ 0.