the area of a trapezoid is 32 square meters. if one base is 6 meters longer than the height and the other base is 2 meters longer than the height. find the height of the trapezoid

Answer 1

Let's start with what we know:

area of a trapezoid = (base1 plus base2) divided by 2 times height or a = (b1 + b2) / 2 * h

base1 = height (h) + 6

base2 = height (h) + 2

a = 32 square meters

So now we start replacing values with givens in order to solve the equation for height, our only unknown.

a = (b1 + b2) / 2 * h

32 = (h+6+h+2) / 2 * h

32 = (2h + 8)/2 * h

32 = (h + 4) * h

32 = h squared + 4h

h = 4 meters

base 1 = 4 + 6 = 10 meters

base2 = 4 + 2 = 6 meters

I wrote the solution for h in bold italics because I solved (32 = h squared + 4h) by thinking about the numbers themselves. This means I assume that we are working with positive whole numbers. There are only 5 possible values for h {1, 2, 3, 4, 5}, zero won't help us. I stopped my list at 5 because 5 * 5 + 4 * 5 is 25 + 20 = 40 (too big). The next logical value to test is 4, which works: 4 * 4 + 4 * 4 = 16 + 16 = 32 (just right).

double check when h = 4:

32 = (10+6) / 2 * 4

32 = 16 / 2 * 4

32 = 8 * 4

32 = 32 👍

HOWEVER, if we want to play by the rules, there are 2 possible answers to the equation:

...

32 = h squared + 4h

0 = h squared + 4h - 32 Quadratic 😯

0 = (h + 8)(h - 4)

h = 4, h = -8

double check where h = -8:

base 1 = -8 + 6 = -2 meters

base2 = -8 + 2 = -6 meters

32 = (-2 + -6) / 2 * -8

32 = -8 / 2 * -8

32 = -4 * -8

32 = 32 👍

You CAN have a negative measurement, it just puts the figure in a different quadrant on an x-y plot. Of course, we don't usually model the real world in negative measurements.

Now you will have to decide based on what you are learning in class. Will you use the simple h=4 version, or the correct h= -8, h=4?

Answer 2

Area of a trapezoid = [ (length of base a + length of base b) / 2 ] * height of the trapezoid

h = height of the trapezoid

base a = h + 6

base b = h + 2

Area = 32

Using our formula for the area of a trapezoid:

32 = [ (h+2 + h+6) / 2] * h

32 = [ (2h+8) / 2] * h

64 = [2h+8] * h

64 = 2h² + 8h

0 = 2h² + 8h - 64

0 = h² +4h - 32

0 = (h+8) (h-4)

h= -8 or h= 4

The height cannot be negative; therefore the height of the trapezoid is 4.