Question

A landscaper is designing a flower garden in the shape of a trapezoid. She wants the length of the shorter base to be 3 yards greater than the height, and the length of the longer base to be 7 yards greater than the height. She wants the area to be 225 square yards. The situation is modeled by the equation 1/2h(b1+b2), where h is height in yards, and b1 and b2 are the length of the bases in yards. Complete the square to find the height that will give the desired area.

Answer 1

@Hero can you help me out again? I'm stuck on this problem... I don't know how to set up the equation

Answer 2

Okay, basically, the best way to approach this is to start with the formula

\[A_{\text{trap}} = \frac{(b_1 + b_2)h}{2}\]

Answer 3

okay

Answer 4

Then write \(b_1\) and \(b_2\) in terms of the height. Try doing so.

Answer 5

are they x+3 and x+7?

Answer 6

Close:
\(b_1 = h + 3\)
\(b_2 = h + 7\)

Answer 7

oooh okay. So now put those into the equation, correct?

Answer 8

Yes

Answer 9

\[A=\frac{ (h+3+h+7)h }{ 2 }\]

Answer 10

Looks good so far

Answer 11

\[A=\frac{ 2h+10)h }{ 2 }\]

Answer 12

oops i forgot the parenthesis

Answer 13

You already know the area as well so we can insert that as well.

Answer 14

oh right! okay
\[225=\frac{ (2h+10)h }{ 2 }\]

Answer 15

Okay, now try solving for h

Answer 16

so i distribute the h first, right?

Answer 17

You can do that, but first I would recommend getting rid of that denominator. Multiply both sides by two.

Answer 18

okay
\[450=(2h+10)h\]

Answer 19

now distribute the h, correct?

Answer 20

\[450=2h ^{2} +10h\]

Answer 21

That looks good so far. What can you do next? Do you know?

Answer 22

no..

Answer 23

factor maybe?

Answer 24

Well, it would be a good idea to make sure what you have is in the absolute simplest form first before trying to factor anything.

Answer 25

We can divide both sides by a number to reduce the equation, correct?

Answer 26

oh! 2?

Answer 27

Yup

Answer 28

\[225=x ^{2}+5\]

Answer 29

i meant 5x

Answer 30

or h hahaha

Answer 31

You shouldn't change the \(h's\) to \(x's\)

Answer 32

Yes try to keep track of the variables and other things

Answer 33

yeah, sorry I'm just used to x, my mistake

Answer 34

It's okay. It happens.

Answer 35

so now we factor?

Answer 36

So we have \(225 = h^2 + 5h\)

Answer 37

Right but we have to put it in an appropriate form first before factoring. Or perhaps at this point they want you to complete the square

Answer 38

225=(x+5)(x+0)

Answer 39

oh...so how do we do that?

Answer 40

Basically, when you have a quadratic equation of the form

\(x^2 + bx = c\)

You have to add\(\left(\dfrac{b}{2}\right)^2\) to both sides.

In this case, b = 5 and c = 225

Answer 41

okay....

Answer 42

Sorry, you kind of lost me

Answer 43

To complete the square, you simply add \(\left(\dfrac{b}{2}\right)^2\) to both sides. In this case, b = 5 to add \(\left(\dfrac{5}{2}\right)^2\) to both sides.

Answer 44

ooooh okay

Answer 45

so now the equation would be

Answer 46

so now the equation would be\[(\frac{ 5 }{ 2 })^{2}\times225=(\frac{ 5 }{ 2 })x^{2}+5x\]

Answer 47

you'rere supposed to \(\color\red{\text{ADD}}\) \(\left(\dfrac{5}{2}\right)^2\) to both sides.

Answer 48

not multiply