contestada

Adah wants to buy a rug for a room that is 19 feet wide and 26 feet long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 330 square feet of carpeting. What dimensions should the rug have?

❎ No Spam ​

Respuesta :

Starrysoul100

[tex]\bold{\huge{\underline{ Solution }}}[/tex]

Given :-

  • Adah wants to buy a rug for room that is 19 feet wide and 26 feet long.
  • She wants to leave a uniform strip of floor around the rug
  • She can only afford to buy 330 sq. feet carpet.

To Find :-

  • We have to find the dimensions of the rug .

Let's Begin :-

Here, we have

  • The dimensions of Adah room that is 19 feet wide and 26 feet long.

But,

  • She wants to leave a uniform strip of floor around the rug.

Therefore,

Let the width of the uniform strip of the floor be x

So, The new dimensions of the room will be

  • [tex]\sf{ (19 - 2x)\: \:and \: \: (26 - 2x)}[/tex]
  • She can only afford to buy 330 sq.feet

We know that,

Area of rectangle

[tex]\bold{\red{ = Length} }{\bold{\red{\times{ Breath}}}}[/tex]

Subsitute the required values,

[tex]\sf{ ( 19 - 2x) }{\sf{\times{(26 - 2x) = 330}}}[/tex]

[tex]\sf{19(26 - 2x) }{\sf{\times{ - 2x (26 - 2x) = 330}}}[/tex]

[tex]\sf{ 494 - 38x - 52x + 4x^{2} = 330 }[/tex]

[tex]\sf{ 494 - 38x - 52x + 4x^{2} = 330 }[/tex]

[tex]\sf{ - 90x + 4x^{2} = 330 - 494 }[/tex]

[tex]\sf{ - 90x + 4x^{2} = - 164 }[/tex]

  • Arrange the given terms in general quadratic equation that ax² + bx + c = 0

[tex]\sf{ 4x^{2} - 90x + 164 = 0 }[/tex]

[tex]\sf{ 4x^{2} - 82x - 8x + 164 = 0 }[/tex]

[tex]\sf{ 2x( 2x - 41 ) -4( 2x - 41 ) = 0 }[/tex]

[tex]\sf{ (2x - 4) ( 2x - 41) = 0}[/tex]

[tex]\sf{ x = 2 , x = }{\dfrac{41}{2}}[/tex]

  • We can only take only 1 value that is x = 2 because 41/2 will give negative result and dimensions of rectangle can never be negative.

Therefore,

Length of the rug

[tex]\sf{ (19 - 2x) = 19 - 2(2)}[/tex]

[tex]\sf{ = 19 - 4}[/tex]

[tex]\sf{ = 15\: feet}[/tex]

Breath of the rug

[tex]\sf{ (26 - 2x) = 19 - 2(2)}[/tex]

[tex]\sf{ = 26 - 4}[/tex]

[tex]\sf{ = 22\:feet}[/tex]

Hence, The dimensions of the rug are 15 feet and 22 feet.

Otras preguntas