Answer
Verified
396k+ views
Hint: Here, we apply the formula for the area as well as the perimeter of a rectangle to find the length and breadth of the rectangle and then we will apply the Pythagoras theorem to find the length of the diagonal of the rectangle.
The area of the rectangle is given by:-
\[area = length \times breadth\]
The perimeter of the rectangle is given by:
\[perimeter = 2(length + breadth)\]
The Pythagoras theorem states that the sum of squares of the base and height of a triangle is equal to the square of its hypotenuse.
\[{\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {height} \right)^2}\]
Complete step by step solution:
It is given that area of the rectangle is \[120{m^2}\]
The perimeter of the rectangle is $46m$.
Let ABCD be the rectangle with length=\[l\] and breadth= \[b\]
Then, the area of the rectangle is given by:-
\[area = length \times breadth\]
Therefore putting in the values we get:-
\[120 = l \times b.....................\left( 1 \right)\]
Also, the perimeter of the rectangle is given by:
\[perimeter = 2(length + breadth)\]
Putting in the values we get:-
\[ 46 = 2\left( {l + b} \right) \]
$ l + b = \dfrac{{46}}{2} $
On simplification,
$ l + b = 23.............\left( 2 \right) $
Now solving equations 1 and 2 we get:-
From equation 2 we get:-
\[b = 23 - l.....................\left( 3 \right)\]
Putting this value in equation 1 we get:-
\[
l\left( {23 - l} \right) = 120 \\
23l - {l^2} = 120 \\
{l^2} - 23l + 120 = 0 \\
\]
Solving this quadratic equation by middle term split we get:-
\[ {l^2} - 15l - 8l + 120 = 0 \]
\[ l\left( {l - 15} \right) - 8\left( {l - 15} \right) = 0 \]
on simplification,
\[ \left( {l - 15} \right)\left( {l - 8} \right) = 0 \]
\[ l = 15;\, l = 8 \]
Putting the value of $l$ back in equation 3 we get:-
When $l=15m$
Then,
\[b = 23 - 15 \]
$\Rightarrow b = 8 $
When $l=8m$
\[\Rightarrow b = 23 - 8 \]
$\Rightarrow b = 15$
Now applying Pythagoras theorem in \[\Delta ABC\] we get:-
Since the formula of Pythagoras is given by:-
\[{\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {height} \right)^2}\]
Therefore,
\[{\left( {AC} \right)^2} = {\left( {AB} \right)^2} + {\left( {BC} \right)^2}\]
Which implies:
\[{\left( {diagonal} \right)^2} = {\left( {length} \right)^2} + {\left( {breadth} \right)^2}\]
Now putting in the values we get:-
\[\Rightarrow {\left( {diagonal} \right)^2} = {\left( 8 \right)^2} + {\left( {15} \right)^2} \]
$\Rightarrow {\left( {diagonal} \right)^2} = 64 + 225$
$\Rightarrow {\left( {diagonal} \right)^2} = 289 $
$\Rightarrow diagonal = \sqrt {289} $
$\Rightarrow diagonal = 17m $
$\therefore$ The length of the diagonal is 17 meter. Hence, option (C) is the correct answer.
Note:
The student can also apply quadratic formula to solve the quadratic equation and find the value of \[l\]
For any quadratic equation of the form \[a{x^2} + bx + c = 0\]
The roots of the equation are given by:-
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Hence applying this formula for the equation
\[{l^2} - 23l + 120 = 0\]
We get:-
\[
l = \dfrac{{ - \left( { - 23} \right) \pm \sqrt {{{\left( { - 23} \right)}^2} - 4\left( 1 \right)\left( {120} \right)} }}{{2\left( 1 \right)}} \\
l = \dfrac{{23 \pm \sqrt {529 - 480} }}{2} \\
l = \dfrac{{23 \pm \sqrt {49} }}{2} \\
l = \dfrac{{23 \pm 7}}{2} \\
l = \dfrac{{23 + 7}}{2}or{\text{ }}l = \dfrac{{23 - 7}}{2} \\
l = \dfrac{{30}}{2}or{\text{ }}l = \dfrac{{16}}{2} \\
l = 15cm{\text{ }}or{\text{ }}l = 8cm \\
\].
The area of the rectangle is given by:-
\[area = length \times breadth\]
The perimeter of the rectangle is given by:
\[perimeter = 2(length + breadth)\]
The Pythagoras theorem states that the sum of squares of the base and height of a triangle is equal to the square of its hypotenuse.
\[{\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {height} \right)^2}\]
Complete step by step solution:
It is given that area of the rectangle is \[120{m^2}\]
The perimeter of the rectangle is $46m$.
Let ABCD be the rectangle with length=\[l\] and breadth= \[b\]
Then, the area of the rectangle is given by:-
\[area = length \times breadth\]
Therefore putting in the values we get:-
\[120 = l \times b.....................\left( 1 \right)\]
Also, the perimeter of the rectangle is given by:
\[perimeter = 2(length + breadth)\]
Putting in the values we get:-
\[ 46 = 2\left( {l + b} \right) \]
$ l + b = \dfrac{{46}}{2} $
On simplification,
$ l + b = 23.............\left( 2 \right) $
Now solving equations 1 and 2 we get:-
From equation 2 we get:-
\[b = 23 - l.....................\left( 3 \right)\]
Putting this value in equation 1 we get:-
\[
l\left( {23 - l} \right) = 120 \\
23l - {l^2} = 120 \\
{l^2} - 23l + 120 = 0 \\
\]
Solving this quadratic equation by middle term split we get:-
\[ {l^2} - 15l - 8l + 120 = 0 \]
\[ l\left( {l - 15} \right) - 8\left( {l - 15} \right) = 0 \]
on simplification,
\[ \left( {l - 15} \right)\left( {l - 8} \right) = 0 \]
\[ l = 15;\, l = 8 \]
Putting the value of $l$ back in equation 3 we get:-
When $l=15m$
Then,
\[b = 23 - 15 \]
$\Rightarrow b = 8 $
When $l=8m$
\[\Rightarrow b = 23 - 8 \]
$\Rightarrow b = 15$
Now applying Pythagoras theorem in \[\Delta ABC\] we get:-
Since the formula of Pythagoras is given by:-
\[{\left( {hypotenuse} \right)^2} = {\left( {base} \right)^2} + {\left( {height} \right)^2}\]
Therefore,
\[{\left( {AC} \right)^2} = {\left( {AB} \right)^2} + {\left( {BC} \right)^2}\]
Which implies:
\[{\left( {diagonal} \right)^2} = {\left( {length} \right)^2} + {\left( {breadth} \right)^2}\]
Now putting in the values we get:-
\[\Rightarrow {\left( {diagonal} \right)^2} = {\left( 8 \right)^2} + {\left( {15} \right)^2} \]
$\Rightarrow {\left( {diagonal} \right)^2} = 64 + 225$
$\Rightarrow {\left( {diagonal} \right)^2} = 289 $
$\Rightarrow diagonal = \sqrt {289} $
$\Rightarrow diagonal = 17m $
$\therefore$ The length of the diagonal is 17 meter. Hence, option (C) is the correct answer.
Note:
The student can also apply quadratic formula to solve the quadratic equation and find the value of \[l\]
For any quadratic equation of the form \[a{x^2} + bx + c = 0\]
The roots of the equation are given by:-
\[x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}\]
Hence applying this formula for the equation
\[{l^2} - 23l + 120 = 0\]
We get:-
\[
l = \dfrac{{ - \left( { - 23} \right) \pm \sqrt {{{\left( { - 23} \right)}^2} - 4\left( 1 \right)\left( {120} \right)} }}{{2\left( 1 \right)}} \\
l = \dfrac{{23 \pm \sqrt {529 - 480} }}{2} \\
l = \dfrac{{23 \pm \sqrt {49} }}{2} \\
l = \dfrac{{23 \pm 7}}{2} \\
l = \dfrac{{23 + 7}}{2}or{\text{ }}l = \dfrac{{23 - 7}}{2} \\
l = \dfrac{{30}}{2}or{\text{ }}l = \dfrac{{16}}{2} \\
l = 15cm{\text{ }}or{\text{ }}l = 8cm \\
\].
Recently Updated Pages
The branch of science which deals with nature and natural class 10 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
Define absolute refractive index of a medium
Find out what do the algal bloom and redtides sign class 10 biology CBSE
Prove that the function fleft x right xn is continuous class 12 maths CBSE
Find the values of other five trigonometric functions class 10 maths CBSE
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Two charges are placed at a certain distance apart class 12 physics CBSE
Difference Between Plant Cell and Animal Cell
What organs are located on the left side of your body class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
The planet nearest to earth is A Mercury B Venus C class 6 social science CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
What is BLO What is the full form of BLO class 8 social science CBSE