Asked by Anne Marie Kratz on May 11, 2024
Verified
A rectangular playing field with a perimeter of 120120120 meters is to have an area of at least 495495495 square meters. Within what bounds must the length of the field lie?
A) between 303030 and 30+9530 + 9 \sqrt { 5 }30+95 meters
B) between 30−9530 - 9 \sqrt { 5 }30−95 and 30+9530 + 9 \sqrt { 5 }30+95 meters
C) between 281228 \frac { 1 } { 2 }2821 and 303030 meters
D) between 281228 \frac { 1 } { 2 }2821 and 30+9530 + 9 \sqrt { 5 }30+95 meters
E) between 281228 \frac { 1 } { 2 }2821 and 999 meters
Perimeter
The perimeter of a shape is the total length of its boundary, measured along the outer sides.
Rectangular Playing Field
A flat area of land designated for sports or games, with a length typically greater than its width, bounded by straight edges at right angles.
Square Meters
A unit of area measurement in the metric system, equivalent to the area of a square with sides one meter in length.
- Understand the relationship between perimeter and area of a rectangle and apply this understanding to solve inequality problems related to dimensions of geometric shapes.
- Resolve inequalities that incorporate algebraic expressions.
- Apply mathematical reasoning to determine the bounds within which a geometric dimension must lie.
Verified Answer
JT
Jonathan TheodoreMay 15, 2024
Final Answer :
B
Explanation :
Let the length be lll and the width be www . The perimeter is 2l+2w=1202l + 2w = 1202l+2w=120 , so l+w=60l + w = 60l+w=60 . The area is lw≥495lw \geq 495lw≥495 . Solving the system, we find the bounds for lll are between 30−9530 - 9 \sqrt { 5 }30−95 and 30+9530 + 9 \sqrt { 5 }30+95 .
Learning Objectives
- Understand the relationship between perimeter and area of a rectangle and apply this understanding to solve inequality problems related to dimensions of geometric shapes.
- Resolve inequalities that incorporate algebraic expressions.
- Apply mathematical reasoning to determine the bounds within which a geometric dimension must lie.