Question

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?

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Answer 1

`\bold{\huge{\underline{ Solution }}}`

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

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

We know that,

Area of rectangle

`\bold{\red{ = Length} }{\bold{\red{*{ Breath}}}}`

Subsitute the required values,

`\sf{ ( 19 - 2x) }{\sf{*{(26 - 2x) = 330}}}`

`\sf{19(26 - 2x) }{\sf{*{ - 2x (26 - 2x) = 330}}}`

`\sf{ 494 - 38x - 52x + 4x^(2) = 330 }`

`\sf{ 494 - 38x - 52x + 4x^(2) = 330 }`

`\sf{ - 90x + 4x^(2) = 330 - 494 }`

`\sf{ - 90x + 4x^(2) = - 164 }`

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

`\sf{ 4x^(2) - 90x + 164 = 0 }`

`\sf{ 4x^(2) - 82x - 8x + 164 = 0 }`

`\sf{ 2x( 2x - 41 ) -4( 2x - 41 ) = 0 }`

`\sf{ (2x - 4) ( 2x - 41) = 0}`

`\sf{ x = 2 , x = }{(41)/(2)}`

• 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

`\sf{ (19 - 2x) = 19 - 2(2)}`

`\sf{ = 19 - 4}`

`\sf{ = 15\: feet}`

Breath of the rug

`\sf{ (26 - 2x) = 19 - 2(2)}`

`\sf{ = 26 - 4}`

`\sf{ = 22\:feet}`

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