Respuesta :
Answer: The dimensions are: 5.7 ft. by 3 ft. .
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(or, write as: The length is 5.7 ft. The width is 3 ft.). {Note: Do not forget to include the "units", which are "feet", or "ft.", in this case!}.
________________
Explanation:
___________________
Consider a flag to be a "rectangle". We are asked to find the dimensions; which are the "length" and the "width". Let "l" represent the length; and "w" represent the "width". The "area", or "A", of a rectangle = the length multiplied by the width; or: "A = l * w " .
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So, given: "the length of a flag is 0.3 foot less than twice its width" ;
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We express this as: " l = 2w - 0.3 " .
____________________________________________________
Now, note that the Perimeter, "P", of the rectangle:
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" P = 2l + 2w " ;
____________________________________________________
We are given: "the perimeter is 14.4 feet longer than the width" ;
____________________________________________________
We express this as: "P = 14.4 + w " ;
____________________________________________________
so: P = 2l + 2w = 14.4 + w ;
____________________________________________________
then: 2l + 2w = 14.4 + w .
____________________________________________________
Since: " l = 2w - 0.3" ; we can substitute this value, "(2w - 0.3)" , for the "l" in the above equation; and solve for "w" ;
____________________________________________________
→ 2l + 2w = 14.4 + w ;
____________________________________________________
→ 2(2w - 0.3) + 2w = 14.4 + w ;
____________________________________________________
Now, subtract "w" from EACH SIDE of the equation:
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2(2w - 0.3) + 2w - w = 14.4 + w - w ;
____________________________________________________
to get: 2(2w - 0.3) + w = 14.4;
_______________________________
Note the distributive property of multiplication:
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a(b+c) = ab + ac ; AND:
a(b -c) = ab - ac ;
__________________________
So; 2(2w - 0.3) = (2*2w) - (2* 0.3) = 4w - 0.6 ;
___________________________________________
Rewrite the equation, substituting: "4w - 0.6" in lieu of: "2(2w - 0.3)" ;
______________________________________________________
→ 2(2w - 0.3) + w = 14.4 ;
→ Rewrite as: 4w - 0.6 + w = 14.4 ;
________________ ___________________________________
Now, combine the "like terms" that appear on the left-hand side of the equation: +4w + w = 5w ; and rewrite the equation:
____________________________________________________
→ 5w - 0.6 = 14.4 ;
____________________________________________________
→ Now, add "0.6" to EACH side of the equation:
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→ 5w - 0.6 + 0.6 = 14.4 + 0.6 ;
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to get: 5w = 15 ;
____________________________________________________
Now, divide EACH side of the equation by "5" ; to isolate "w" on one side of the equation; and to solve for "w" (the width, which is one of the dimensions);
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→ 5w / 5 = 15/ 5 ;
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→ w = 3 ;
_________________________________________
Now, we need to find the length, "l".
___________________________________
Since, " l = 2w - 0.3 ; we can substitute our known value of "w", which is "3", into the equation, and solve for "l" ;
______________________________________________
→ l = (2*3) - 0.3 = 6 - 0.3 = 5.7 ;
→ l = 5.7 .
_____________________________________________
Now, to check our work, does:
___________________________
2l + 2w = 14.4 + w ?
_______________________________________________
Let us plug our solved values, "5.7" for "l", and "3" for "w"; so see if the equation holds true.
___________________________________________________
Let us start with the right side of the equation:
___________________________________________________
14.4 + w ; 14.4 + w = 14.4 + 3 = 17.4.
___________________________________________________
Now, let us consider the left side of of the equation:
___________________________________________________
2l + 2w = (2*5.7) + (2*3) = 11.4 + 6 = 17.4 .
___________________________________________________
17.4 =? 17.4 ? Yes!.
___________________________________________________
So, the answer is: The dimensions are: 5.7 ft. by 3 ft. .
(or, write as: The length is 5.7 ft. The width is 3 ft.).
{Note: Do not forget to include the "units", which are "feet", or "ft.", in this case!}.
__________________________________________________
_________________________________________________
(or, write as: The length is 5.7 ft. The width is 3 ft.). {Note: Do not forget to include the "units", which are "feet", or "ft.", in this case!}.
________________
Explanation:
___________________
Consider a flag to be a "rectangle". We are asked to find the dimensions; which are the "length" and the "width". Let "l" represent the length; and "w" represent the "width". The "area", or "A", of a rectangle = the length multiplied by the width; or: "A = l * w " .
____________________________________________________
So, given: "the length of a flag is 0.3 foot less than twice its width" ;
____________________________________________________
We express this as: " l = 2w - 0.3 " .
____________________________________________________
Now, note that the Perimeter, "P", of the rectangle:
____________________________________________________
" P = 2l + 2w " ;
____________________________________________________
We are given: "the perimeter is 14.4 feet longer than the width" ;
____________________________________________________
We express this as: "P = 14.4 + w " ;
____________________________________________________
so: P = 2l + 2w = 14.4 + w ;
____________________________________________________
then: 2l + 2w = 14.4 + w .
____________________________________________________
Since: " l = 2w - 0.3" ; we can substitute this value, "(2w - 0.3)" , for the "l" in the above equation; and solve for "w" ;
____________________________________________________
→ 2l + 2w = 14.4 + w ;
____________________________________________________
→ 2(2w - 0.3) + 2w = 14.4 + w ;
____________________________________________________
Now, subtract "w" from EACH SIDE of the equation:
_____________________________________________________
2(2w - 0.3) + 2w - w = 14.4 + w - w ;
____________________________________________________
to get: 2(2w - 0.3) + w = 14.4;
_______________________________
Note the distributive property of multiplication:
__________________________________
a(b+c) = ab + ac ; AND:
a(b -c) = ab - ac ;
__________________________
So; 2(2w - 0.3) = (2*2w) - (2* 0.3) = 4w - 0.6 ;
___________________________________________
Rewrite the equation, substituting: "4w - 0.6" in lieu of: "2(2w - 0.3)" ;
______________________________________________________
→ 2(2w - 0.3) + w = 14.4 ;
→ Rewrite as: 4w - 0.6 + w = 14.4 ;
________________ ___________________________________
Now, combine the "like terms" that appear on the left-hand side of the equation: +4w + w = 5w ; and rewrite the equation:
____________________________________________________
→ 5w - 0.6 = 14.4 ;
____________________________________________________
→ Now, add "0.6" to EACH side of the equation:
____________________________________________________
→ 5w - 0.6 + 0.6 = 14.4 + 0.6 ;
____________________________________________________
to get: 5w = 15 ;
____________________________________________________
Now, divide EACH side of the equation by "5" ; to isolate "w" on one side of the equation; and to solve for "w" (the width, which is one of the dimensions);
____________________________________________________
→ 5w / 5 = 15/ 5 ;
____________________________________________________
→ w = 3 ;
_________________________________________
Now, we need to find the length, "l".
___________________________________
Since, " l = 2w - 0.3 ; we can substitute our known value of "w", which is "3", into the equation, and solve for "l" ;
______________________________________________
→ l = (2*3) - 0.3 = 6 - 0.3 = 5.7 ;
→ l = 5.7 .
_____________________________________________
Now, to check our work, does:
___________________________
2l + 2w = 14.4 + w ?
_______________________________________________
Let us plug our solved values, "5.7" for "l", and "3" for "w"; so see if the equation holds true.
___________________________________________________
Let us start with the right side of the equation:
___________________________________________________
14.4 + w ; 14.4 + w = 14.4 + 3 = 17.4.
___________________________________________________
Now, let us consider the left side of of the equation:
___________________________________________________
2l + 2w = (2*5.7) + (2*3) = 11.4 + 6 = 17.4 .
___________________________________________________
17.4 =? 17.4 ? Yes!.
___________________________________________________
So, the answer is: The dimensions are: 5.7 ft. by 3 ft. .
(or, write as: The length is 5.7 ft. The width is 3 ft.).
{Note: Do not forget to include the "units", which are "feet", or "ft.", in this case!}.
__________________________________________________
