Quadratic Equation: A new build estate requires a concrete path to be laid around....

[JimCrown]

Hi, I have been stumped on this question but, feel like I may have the right answer. Any help would be greatly appreciated.

A new build estate requires a concrete path to be laid around the edge of each back garden. The first group of houses to be built have a garden 8m wide and 10m long. The path has constant width and is laid around the edge of the garden. If the area of the path is 100m2, calculate, by deriving a quadratic equation the width of the path (w) Find a quadratic expression to calculate the width of the path hence find the width using the known values. Create a spreadsheet to check your calculations and find an expression that could be used for gardens of different areas.

Working out:

8 x 10 = 80m^2
8+2x multiplied by 10+2x minus 80
(8+2x)(10+2x) - 80 = 100m^2
80 + 16x = 20x + 4x^2 - 80 = 100m^2
36x + 4x^2 = 100m^2

x^2 + 9x - 25 = 0

-b +/- √b^2 - 4ac
______
2a

-9 +/- √81-4(-25)
______
-50

-9 +/- √81+100
______
-50

-9 +/- √81-4x(x-25)

-9 +/- √181
______
2
x= -9 1
____ + ____ √181 = 2.22681202 metres for the width of the path

 

[JimCrown]

Anybody my friend got the answer of 2.4 (he rounded his answer up)

 

[stapel]

A new build estate requires a concrete path to be laid around the edge of each back garden. The first group of houses to be built have a garden 8m wide and 10m long. The path has constant width and is laid around the edge of the garden. If the area of the path is 100m2, calculate, by deriving a quadratic equation the width of the path (w) Find a quadratic expression to calculate the width of the path hence find the width using the known values. Create a spreadsheet to check your calculations and find an expression that could be used for gardens of different areas.

Working out:

8 x 10 = 80m^2
8+2x multiplied by 10+2x minus 80
(8+2x)(10+2x) - 80 = 100m^2
80 + 16x = 20x + 4x^2 - 80 = 100m^2
36x + 4x^2 = 100m^2

x^2 + 9x - 25 = 0

-b +/- √b^2 - 4ac
______
2a

-9 +/- √81-4(-25)
______
-50

-9 +/- √81+100
______
-50

-9 +/- √81-4x(x-25)

-9 +/- √181
______
2
x= -9 1
____ + ____ √181 = 2.22681202 metres for the width of the path

I think your working, after "x² + 9x - 25 = 0", is meant to be as follows:

. . . . .\(\displaystyle x\, =\, \dfrac{-b\, \pm\, \sqrt{\strut b^2\, -\, 4ac\,}}{2a}\)

. . . . .\(\displaystyle x\, =\, \dfrac{-(9)\, \pm\, \sqrt{\strut (9)^2\, -\, 4(1)(-25)\,}}{2(1)}\)

. . . . .\(\displaystyle x\, =\, \dfrac{-9\, \pm\, \sqrt{\strut 81\, +\, 100\,}}{2}\)

. . . . .\(\displaystyle x\, =\, \dfrac{-9\, \pm\, \sqrt{\strut 181\,}}{2}\)

I'm afraid I don't know where you got a "-50" for the denominator, and I can't follow your work from "-9 +/- √81-4x(x-25)" onwards. Sorry.

Where are you stuck in the process of using a spreadsheet, as instructed, to check your work? Please be specific. Thank you!

 

[Ishuda]

Anybody my friend got the answer of 2.4 (he rounded his answer up)

I got the answer you did which was the positive solution to
4x² + 36x = 100
or, equivalently,
x² + 9x = 25.

 

[lookagain]

Working out:

8 x 10 = 80m^2\(\displaystyle \ \ \ \ \ \ \ \ \ \)Avoid the "x" symbol for multiplication in these algebra problems.
8+2x multiplied by 10+2x minus 80
(8+2x)(10+2x) - 80 = 100m^2
80 + 16x = 20x + 4x^2 - 80 = 100m^2\(\displaystyle \ \ \ \ \ \ \ \ \)The first equals sign is supposed to be an addition sign.

\(\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)Just keep it simple. Keep the units out of the equation, else you'll need them on
both sides of the equation.

36x + 4x^2 = 100m^2

x^2 + 9x - 25 = 0

-b +/- √b^2 - 4ac
______
2a

-9 +/- √81-4(-25)
______\(\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)This denominator is wrong, and it's fudging to go from here to ...
-50

-9 +/- √81+100
______
-50

-9 +/- √81-4x(x-25)

-9 +/- √181
______\(\displaystyle \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \)here, where the denominator is correct.
2
x= -9 1
____ + ____ √181 = 2.22681202 metres for the width of the path

Aside from not defining your variable, not grouping the terms of your radicands,
being inconsistent with your units on both sides of your equations, your main
problem is stumbling with the use of the quadratic formula.

- - - - -- - - - - - - - - -- - - - - - - - - - - -- -- - - - -- - - - -- - - - - -- -- - - -


8*10 = 80

(8+2x) multiplied by (10+2x) minus 80

(8+2x)(10+2x) - 80 = 100

80 + 16x + 20x + 4x^2 - 80 = 100

36x + 4x^2 = 100

4x^2 + 36x - 100 = 0

x^2 + 9x - 25 = 0


-b +/- √(b^2 - 4ac)
________________ = x
. . . . . . . 2a

-9 +/- √[81 - 4(1)(-25)]
____________________ = x
. . . . . . . . 2(1)

-9 +/- √(81+100)
_______________ = x
. . . . . . 2


-9 +/- √(181)
____________ = x
. . . . . 2


x =

-9 + √(181)
__________
. . . . 2


x ~ 2.2268


The width of the path is about 2.2268 m.


___________________________________________


JimCrown, as far as your friend goes, the value rounds to 2.2, not 2.4.