Dimensions, Height- Yard, Wall

[Guest]

A wall of a certain height is constructed around a yard. The length of one side of the yard is 2 yards less than 8 times the height of the wall, and the length of the adjacent side is 5 yards less than 6 times the height of the wall. The number of square yards in the lawn is greater than the number in the wall by 178. Determine dimensions of the yard, and height of wall.

My partial attempt:

Let H = height of wall.

Let A = one side of yard.

Let B = other side of yard.

Let L = square yards in lawn.

Let W = square yards in wall.

A = 8H - 2

B = 6H - 5

L = W + 178

How do I proceed from this point ?

[Guest]

"The number of square yards in the lawn"....I assume you mean "The number of square yards in the lawn"

Side A = 8H - 2
Side B = 6H - 5

Area of yard = SideA x SideB = (8H - 2)(6H - 5)

Area of wall = Height x Perimeter = = Height x 2 x (SideA + SideB)) = 2H(14H - 7)

Area of yard = Area of wall + 178

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

Solve for H................then work out SideA and SideB from above eqns.............

[Guest]

The number of square yards in the lawn"....I assume you mean "The number of square yards in the yard"

[Guest]

The number of square yards in the lawn"....I assume you mean "The number of square yards in the yard" Yes.

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H + 40H + 12H + 10 = 28H - 14H + 178 (multiplying out)

100H + 10 - 10 - 14H = 14H + 178 - 10 - 14H (combining like terms)

86H = 168

86H/86 = 168 / 86

H = 1.95

Side A - 8(1.95) - 2

15.6 - 2 = 13.6

Side B - 6(1.95) - 5
11.7 - 5 = 6.7

13.6 x 6.7 = 91.12 = area of yard.

?? I am not sure.

[Guest]

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H + 40H + 12H + 10 = 28H - 14H + 178 (multiplying out).....wrong

If you have difficulty doing this line why continue with the other calculations

You are multiplying out the terms in the brackets on each side of the equation...
What is X multiplied by X ?
What is 2X multiplied by 3X ?

[Guest]

"What is X multiplied by X ? X^2 ?
What is 2X multiplied by 3X ?" 6X^2 ?

[Guest]

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H^2 + 40H + 12H + 10 = 28H^2 - 14H + 178 (multiplying out)

48H^2 + 52H + 10 = 28H^2 - 14H + 178

48H^2 + 52H - 14H + 10 - 10 = 28H^2 - 28H^2 +14H - 14H + 178 - 10 (combining terms)

48H^2 - 28H^2 + 38H = 168

20H^2 + 38H = 168

??

[Guest]

"What is X multiplied by X ? X^2 ?......Yes
What is 2X multiplied by 3X ?" 6X^2 ? ......Yes

48H^2 + 40H + 12H + 10 = 28H^2 - 14H + 178 (multiplying out).......Wrong..

Writing down X on its own implies +X as in (plus X).
Writing down -X explicity states it is -X as in (minus X)

What is (+X )multiplied by (-X) ?...
What is (+2X) multiplied by (-5) ?...

[Guest]

"What is (+X )multiplied by (-X) ?... -X^2
What is (+2X) multiplied by (-5) ?..." -10X

[Guest]

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H^2 - 40H -12H -10 = 28H^2 -14H + 178

48H^2 - 28H^2 - 40H + 14H - 12H - 10 + 10 = 178 + 10

20H^2 - 38H = 188

I don't know if that is not correct.

[Guest]

"What is (+X )multiplied by (-X) ?... -X^2..........Yes
What is (+2X) multiplied by (-5) ?..." -10X ..........Yes

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H^2 - 40H -12H -10 = 28H^2 -14H + 178.........Wrong

What is -7 multiplied by -9 ?.....

[Guest]

"What is -7 multiplied by -9 ?....." 63

(8H - 2)(6H - 5) = 2H(14H - 7) + 178

48H^2 - 40H -12H +10 = 28H^2 -14H + 178

48H^2 - 28H^2 - 40H + 14H - 12H - 10 + 10 = 178 - 10

20H^2 - 38H = 168

I don't know.

[Guest]

20H^2 - 38H = 168 .......Yes

I don't know. ........Yes...you are OK so far....This is an ordinary quadratic equation, so solve for H.

In general it will have 2 solutions fo "H".....one will be valid and the other not valid.



For example if we start with an answer of X = something, and work the process in reverse.....we get to a quadratic eqn.


For example.....If.... X = 1 OR X = 2 .....

That is same as X - 1 = 0 OR X - 2 = 0

If we multiply them together....

(X - 1)(X - 2) = 0

and multiply out the brackets....

X^2 - 2X - X + 2 = 0

X^2 - 3X + 2 = 0 .....we get an equation similar to above question

and from this you see if we start with a quadratic, we can work back again and factorise and solve for X.

Not all are as simple as this to factorise.....

But...we can use the quadratic formula to solve for X, OR use completing the square to solve...


So that is all you have to do now.......????

[Guest]

20H^2 - 38H = 168

20H^2 - 38H = 168 -168

20H^2 - 38H - 168 = 0

Using quadratic formula:

-b + - square root b^2 - 4ac / 2a

H = 38 + - sq. rt. 1444 - 4(20)(168) / 40

H = 38 + - sq. rt. 1444 - 4(3360) /40

H = 38 + - sq. rt. 1444 - 13440 /40

H = 38 + - sq. rt. - 11956 / 40

H = 38 + - sq. rt. -109.34 /40

H = 38 + - 2.7335 / 40

H = 1.0183375

That is not correct. I can't solve it.

[Guest]

20H^2 - 38H = 168

20H^2 - 38H = 168 -168....Wrong...You must take away 168 from both sides...20H^2 - 38H -168 = 168 -168......

20H^2 - 38H - 168 = 0 ....You have corrected it here, so its OK from here on...

Using quadratic formula: this is your answer commented on....

H = [-b + - square root (b^2 - 4ac)] / 2a .....use brackets to group things..

you coulde have divided across by 2 to make numbers smaller .....but its OK not to do it...

H = [38 + - sq. rt. (1444 - 4(20)(-168))] / 40 .....168 should be -168 because c = -168....

H = 38 + - sq. rt. 1444 - 4(3360) /40......The above error means everything else below is wrong....
If you correct this you should be able to solve for H OK.....

[Guest]

H = [38 + - sq. rt. (1444 - 4(20)(-168))] / 40

H = 38 + - sq. rt. 1444 - 4 -3360 / 40

H = 38 + - sq. rt. 1444 + 13440 / 40

H = 38 + - sq. rt. 14884 / 40

H = 38 + - 122 / 40

H = 160 / 40

H = 4

Not sure.

[Guest]

You have omitted to keep using the brackets.....so each line does not read correctly....terms not grouped..???
and you only considered the 38+ whatever.... instead of 38+-whatever....

[Guest]

H = [38 + - sq. rt. (1444 - 4 -(3360)] / 40

H = [38 + - sq. rt. (1444 + 13440)] / 40

H = [38 + - sq. rt. 14884] / 40

H = [38 + - 122 ] / 40

H = [38 - 122] / 40

H = -160 / 40

H = -4 Invalid - wall height can't be negative.

[Guest]

H = [38 + - 122 ] / 40 ....Yes OK to here...

Read the + - as Plus OR Minus....Therefore....

H = [38 + 122 ] / 40 .. OR .. H = [38 - 122 ] / 40

So.... H = [160] / 40 .. OR .. H = [-84] / 40

So.... H = 4 .. OR .. H = -2.1

H cannot be negative so H = -2.1 is invalid.

Therefore valid answer is H = 4 yards

Side A = 8H - 2 .... = 32 - 2 = 30 yards
Side B = 6H - 5 .... = 24 - 5 = 21 yards

[Guest]

Thanks.

I have a question:

"Area of wall = Height x Perimeter = = Height x 2 x (SideA + SideB)) = 2H(14H - 7)"

I don't understand the 2H(14H - 7).