Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1999 AMC 8 Problems/Problem 5"
(Added LaTeX)
 
(12 intermediate revisions by 9 users not shown)
Line 1: Line 1:
−
==problem==
+
==Problem==
  
−
A rectangular garden 50 feet long and 10 feet wide is enclosed by a fence. To
+
A rectangular garden 60 feet long and 20 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?
−
make the garden larger, while using the same fence, its shape is changed to a
 
−
square. By how many square feet does this enlarge the garden?
 
−
(A) 100 (B) 200 (C) 300 (D) 400 (E) 500
 
  
−
==solution==
+
<math>\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500</math>
  
−
(D) 400 square feet: The area of the garden was 500
+
==Solution==
−
square feet(50£10) and its perimeter was 120 feet,
+
 
−
2 £ (50 + 10). The square garden is also enclosed
+
We need the same perimeter as a <math>60</math> by <math>20</math> rectangle, but the greatest area we can get. right now the perimeter is <math>160</math>. To get the greatest area while keeping a perimeter of <math>160</math>, the sides should all be <math>40</math>. that means an area of <math>1600</math>. Right now, the area is <math>20 \times 60</math> which is <math>1200</math>. <math>1600-1200=400</math> which is <math>\boxed{D}</math>.
−
by 120 feet of fence so its sides are each 30 feet
+
 
−
long. The square garden's area is 900 square feet
+
==See Also==
−
(30 £ 30). and this has increased the garden area
+
 
−
by 400 square feet.
+
{{AMC8 box|year=1999|num-b=4|num-a=6}}
 +
{{MAA Notice}}

Latest revision as of 13:17, 13 March 2021

Problem

A rectangular garden 60 feet long and 20 feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

$\text{(A)}\ 100 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 300 \qquad \text{(D)}\ 400 \qquad \text{(E)}\ 500$

Solution

We need the same perimeter as a $60$ by $20$ rectangle, but the greatest area we can get. right now the perimeter is $160$. To get the greatest area while keeping a perimeter of $160$, the sides should all be $40$. that means an area of $1600$. Right now, the area is $20 \times 60$ which is $1200$. $1600-1200=400$ which is $\boxed{D}$.

See Also

1999 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 4 		Followed by
Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1999_AMC_8_Problems/Problem_5&oldid=149412"