The perimeter of a rectangle is 210 feet. Describe the possible lengths of a side if the area of the rectangle is not to exceed 1350 square feet.

Question

 I got 90 and 15 as my answers but it told me they were wrong, please help.

Answer 1

perimeter=210 feet

area can't exceed 1350 square feet

possible lengths are: (using whole numbers)

 

1x104

2x103

3x102

4x101

5x100

6x99

7x98

8x97

9x96

10x95

11x94

12x93

13x92

14x91

15x90

16x89=1424 which exceeds 1350 so stop here

 

 

Answer 2

Here's what we know:
1350 = lw (Area is length time width)
210 = 2l + 2w (perimeter is the two length sides and width sides added together)
 
From here, we need to isolate variables. We can do that in the perimeter equation:
210 = 2l + 2w
210 = 2(l + w)
105 = l + w
w = -l + 105
 
Now that we have a value for a variable, we can put that into the area equation:
1350 = l(-l + 105)
1350 = -l2 + 105l
0 = -l2 + 105l - 1350
We'll multiply both sides by -1 to make that -l positive:
0 = l2 - 105l + 1350
 
l = (-(-105) ± √(1052 - 4 * 1 * 1350)) / 2*1
l = (105 ± √(11025 - 5400)) / 2
l = (105 ± √5625) / 2
l = (105 ± 75) / 2
l = 180/2, 30/2
l = 90, 15

Double-checking width: 

w = -90 + 105

w = 15

 
So yeah, I'm coming out to the same answers.

Checking my answers to make sure:

1350 = 90 * 15
1350 = 1350
210 = 2*90 + 2*15
210 = 180 + 30
210 = 210

 

So yes, 90 feet for length and 15 feet for width do make sense. I don't know why you're being told they're wrong.