Dimitri wants to build a rectangular garden in his backyard. He has yet to decide the length, l, and the width, w, but wants the perimeter of the garden to be no less than 40 yards. Which of the following graphs is the solution of the inequality that models the relationship between l and w. (1 point)
Responses
To help Dimitri with his garden, let's first model the perimeter of the rectangular garden with an inequality. The perimeter P of a rectangle is given by:
\[ P = 2l + 2w \]
where \( l \) is the length and \( w \) is the width of the rectangle. Dimitri wants the perimeter to be no less than 40 yards, which gives us the following inequality:
\[ 2l + 2w \geq 40 \]
We can simplify this inequality by dividing every term by 2:
\[ l + w \geq 20 \]
This inequality shows the relationship between the length and width of the garden. We can also express the width \( w \) in terms of the length \( l \), or vice versa, to see what combinations of \( l \) and \( w \) would satisfy the inequality:
\[ w \geq 20 - l \]
Now, to graph this inequality on the coordinate plane where the x-axis represents length (\( l \)) and the y-axis represents width (\( w \)), we can draw the line that represents the boundary case of the inequality:
\[ w = 20 - l \]
This is a straight line with a y-intercept at \( w = 20 \) and an x-intercept at \( l = 20 \). However, since the inequality is \( w \geq 20 - l \), we are interested in the area above this line on the graph—the region that contains all points \( (l, w) \) such that the sum of \( l \) and \( w \) is at least 20, which corresponds to the perimeter being at least 40 yards.
Therefore, the graph of the solution will have shaded area above and including the line \( w = 20 - l \), starting from \( l = 0 \) to some maximum length that Dimitri chooses based on the size of his backyard, and \( w = 0 \) to some maximum width determined by the same constraint.
Since I can't graph it here, imagine a coordinate plane with a line that starts at the point (0, 20) on the y-axis and goes down to the point (20, 0) on the x-axis. The region above and including this line is shaded, and that shaded region represents all the possible lengths and widths for Dimitri's garden that would result in a perimeter of at least 40 yards.
Please note that no actual graph is provided here, but the explanation above should help visualize what the graph should look like.