Open Middle®
Directions: What is the greatest area you can make with a right trapezoid that has a perimeter of 46 units?
Hint
If it wasn’t a trapezoid (any quadrilateral), what would the shape be with the biggest area? What are possible dimensions of a trapezoid that has a perimeter of 46 units? How can we determine their area?
Answer
The right trapezoid with a perimeter of 46 units and the greatest area is one with a height of 12 units, bases of 8 units and of 13 units. That trapezoid has an area of 126 square units. NOTE: this assumes that a trapezoid only has one pair of parallel sides and has integer side lengths.
Source: Patrick McGowan
The right trapezoid with a perimeter of 46 units and greatest area is on earth with height of 12 units, bases of 8 units, and of 13 units. The trapezoid has an area of 126 square units.
The right trapezoid with a perimeter of 46 units and the greatest area is one with a height of 12 units, bases of 8 units and of 13 units. That trapezoid has an area of 126 square units
The height for a 46 unit right trapizode i got is 12 and the bases are 8 and 13 units making int 126 units
The answer I got was, bases of 8 and 13un, and a height of 12un. This added to a perimeter of 46un, and an area of 126un sqared. I got this by messing with numbers untill I found a combination that worked. It took my a while but I found an answer, I checked my answer, and I sucseeded!
The answer that i got is the base is 13 and 8un. with a height of 12 units and the trappizoid has a total area of 126units at first i did all of my cauculation and got 136 but then i checked the answer was incorrect so i tried two find out what i did wrong and after 2 tries i found out what i did wrong.
To get an area of 126 units squared, having a height of 12, with bases of 10 and 11 works. (For the perimeter to be 46, the other leg would be 13 units long.) No one else posted this response, so please explain if this is another possibility or why it wouldn’t work.
base 13 and 8 units height 12 units
The trapezoid has an area 126
We got a trapezoid with b1 = 11, b2 = 11, left side = 23, and rigth side = 1. This gave us a trapezoid with an area of 253 units squared.
With all side lengths being positive integers, the right trapezoid with the greatest area and a perimeter of 46 units is indeed one with a height of 12 units, bases of 8 units and of 13 units. This is because a right trapezoid consists of a rectangle and a right triangle (where all three sides are part of the perimeter). There are only 5 right triangles with integer side lengths smaller base has to be (46 – (3+4+5))/2 = 17 ==> area = (17 + 17+3)*4/2 = 74
8 15 17 ==> smaller base has to be (46 – (8+15+17))/2 = 3 ==> area = (3 + 3+8)*15/2 = 105
6 8 10 ==> smaller base has to be (46 – (6+8+10))/2 = 11 ==> area = (11 + 11+6)*8/2 = 112
9 12 15 ==> smaller base has to be (46 – (9+12+15))/2 = 5 ==> area = (5 + 5+9)*12/2 = 114
5 12 13 ==> smaller base has to be (46 – (5+12+13))/2 = 8 ==> area = (8 + 8+5)*12/2 = 126
If you allow the side lengths to be real numbers, then – since the quadrilateral with the greatest area and a given perimeter is a square and since the square is also a right-angled trapezoid – the solution is a square with side length 11.5 and an area of 132.25 square units.