A rectangular playing field with a perimeter of $120$ meters is to have an area of at least $495$ square meters. Within what bounds must the length of the field lie?

A) between $30$ and $\sqrt { 5 }$ meters
B) between $\sqrt { 5 }$ and $\sqrt { 5 }$ meters
C) between $\frac { 1 } { 2 }$ and $30$ meters
D) between $\frac { 1 } { 2 }$ and $\sqrt { 5 }$ meters
E) between $\frac { 1 } { 2 }$ and $9$ meters

Question

A rectangular playing field with a perimeter of   120 120 120   meters is to have an area of at least   495 495 495   square meters. Within what bounds must the length of the field lie?A) between   30 30 30   and   30 + 9 5 30 + 9 \sqrt { 5 } 30 + 9 5 ​   metersB) between   30 − 9 5 30 - 9 \sqrt { 5 } 30 − 9 5 ​   and   30 + 9 5 30 + 9 \sqrt { 5 } 30 + 9 5 ​   metersC) between   28 1 2 28 \frac { 1 } { 2 } 28 2 1 ​   and   30 30 30   metersD) between   28 1 2 28 \frac { 1 } { 2 } 28 2 1 ​   and   30 + 9 5 30 + 9 \sqrt { 5 } 30 + 9 5 ​   metersE) between   28 1 2 28 \frac { 1 } { 2 } 28 2 1 ​   and   9 9 9   meters

Jonathan Theodore · Accepted Answer

Final Answer : B, Explanation :   Let the length be   l l l   and the width be   w w w  . The perimeter is   2 l + 2 w = 120 2l + 2w = 120 2 l + 2 w = 120  , so   l + w = 60 l + w = 60 l + w = 60  . The area is   l w ≥ 495 lw \geq 495 lw ≥ 495  . Solving the system, we find the bounds for   l l l   are between   30 − 9 5 30 - 9 \sqrt { 5 } 30 − 9 5 ​   and   30 + 9 5 30 + 9 \sqrt { 5 } 30 + 9 5 ​  .

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