To determine which combinations of \( w \) (width) and \( l \) (length) are possible for Mike's fenced-in backyard, we need to check each pair against the two conditions:
- The length must be at least 20 feet: \( l \geq 20 \)
- The total perimeter must not exceed 200 feet: \( 2l + 2w \leq 200 \)
This can be simplified to:
\[ l + w \leq 100 \]
Now, let's check each of the given pairs:
-
\( w = 50 , \text{ft}; , l = 10 , \text{ft} \):
- Check length: \( 10 < 20 \) (does not satisfy)
- Not possible
-
\( w = 10 , \text{ft}; , l = 50 , \text{ft} \):
- Check length: \( 50 \geq 20 \) (satisfies)
- Check perimeter: \( 50 + 10 = 60 \leq 100 \) (satisfies)
- Possible
-
\( w = 20 , \text{ft}; , l = 60 , \text{ft} \):
- Check length: \( 60 \geq 20 \) (satisfies)
- Check perimeter: \( 20 + 60 = 80 \leq 100 \) (satisfies)
- Possible
-
\( w = 90 , \text{ft}; , l = 30 , \text{ft} \):
- Check length: \( 30 \geq 20 \) (satisfies)
- Check perimeter: \( 90 + 30 = 120 \not\leq 100 \) (does not satisfy)
- Not possible
-
\( w = 50 , \text{ft}; , l = 40 , \text{ft} \):
- Check length: \( 40 \geq 20 \) (satisfies)
- Check perimeter: \( 50 + 40 = 90 \leq 100 \) (satisfies)
- Possible
The possible dimensions for Mike’s backyard are:
- \( w = 10 , \text{ft}; , l = 50 , \text{ft} \)
- \( w = 20 , \text{ft}; , l = 60 , \text{ft} \)
- \( w = 50 , \text{ft}; , l = 40 , \text{ft} \)