What is the height of the garden? [Algebra 2: Quadratics]

Question

A landscaper is designing a flower garden in the shape of a trapezoid. She needs the shortest base to be 7 yards greater than the height, and the longer base to be 11 yards greater than the height. She wants the area to be 286 square yards. What is the height of the garden?

Answer 1

A = (0.5)(b₁ + b₂)(h)

286 = (0.5)(h + 11 + h + 7)(h)

572 = (2h + 18)(h)

572 = 2h² + 18h

0 = 2h² + 18h + 572

Can you solve for h and answer?

Answer 2

For this question, you need to know two things:

1. What does the shape of a trapezoid look like
2. The formula to calculate the area of a trapezoid

Coming back to this problem, the question provides us 3 pieces of info.

1. The "shortest base", let's call that b1 is (or equals) 7 yards greater (or add) than the height. b1 = 7 + h.
2. The "longest base", let's call that b2 is 11 yards greater than the height. b2 = 11 + h.
3. The area of the trapezoid is 286.

Combining all the pieces together, we get:

286 = (1/2)( (7+h) + (11+h) )h

Let's work this through step by step making sure to follow the order of operations:

286 = (1/2)( 7 + h + 11 + h)h

286 = (1/2)( 18 + 2h )h

Let's get rid of that (1/2) by multiplying both sides by 2:

286 * 2 = (1/2) * 2 ( 18 + 2h)h

572 = ( 18 + 2h )h

572 = 18h + 2h²

Moving everything to one side and reorganizing it:

0 = 2h² + 18h - 572

Let's make this easier for us, and divide everything by 2 to make the numbers smaller:

0 = h² + 9h - 286

From there, it's an quadratic equation problem, so let's set it up to get started:

0 = (h )(h )

Couple things I can tell from the quadratic question right off the bat.

1. Since "- 286" is negative, the 2 numbers that make up 286, one must be positive and the other must be negative.
2. "+ 9h" is positive, so the positive number must be bigger than the smaller negative number.

Through trial and error, 22 x 13 = 286, and also 22 - 13 = 9

From there, we fill in our quadratic equation:

0 = (h + 22)(h - 13)

Hopefully now, my earlier explanation about which number is positive and negative makes more sense.

The rest should be easy. We get 2 equations from this:

0 = h + 22 0 = h - 13

h = -22 h = 13

Since height cannot negative, h = 13.