Question

This table shows the relationship between area (\[a\]) and width ($w$) for rectangles that have a perimeter of $40$centimeter. Which algebraic equation correctly describes this relationship?

Area (\[a\])${(cm)^2}$	Width ($w$ )$\left( {cm} \right)$
$75$	$5$
$96$	$8$
$100$	$10$
$96$	$12$
$75$	$15$

A)$a = 15w$
B)$a = 10w + 25$
C)$a = 20w - {w^2}$
D)$a = 100 - {w^2}$

Answer 1

Hint: Generally, an algebraic equation is nothing but the method of equating one expression with another expression. That is, the two expressions are said to be equal to each other. Here we don’t need to use the value of the perimeter of a rectangle to solve this question; just the values of width are enough. And our question is to find the correct algebraic equation from the options which matches the given relationship between area and the width of a rectangle.
       We need to just substitute the values of width ($w$ ) in the given algebraic equations. And if the resultant area (\[a\]) is the same as that of the above, then the option is correct.

Complete step by step answer:
I) First, we shall check option A).
     The algebraic expression is $a = 15w$.
    Let us consider the first relationship (i.e.$a = 75,w = 5$)
Substituting the value of $w$ in $a = 15w$, we get
$a = 15 \times 5$
$a = 75$is accepted
    Let us consider the next relationship (i.e.$a = 96,w = 8$)
Substituting the value of$w$ in $a = 15w$, we get
$a = 15 \times 8$
$a = 200 \ne 96$ and it is not applicable.
Hence, option A) is incorrect.
II) Now, we shall check option B).
     The algebraic expression is $a = 10w + 25$.
    Let us consider the first relationship (i.e.$a = 75,w = 5$)
Substituting the value of$w$ in $a = 10w + 25$, we get
$a = 10 \times 5 + 25$
$a = 75$is accepted
    Let us consider the next relationship (i.e.$a = 96,w = 8$)
Substituting the value of$w$ in $a = 10w + 25$, we get
$a = 10 \times 8 + 25$
$a = 105 \ne 96$ and it is not applicable.
Hence, option B) is incorrect.
III) Now, we shall check option C).
     The algebraic expression is $a = 20w - {w^2}$.
    Let us consider the first relationship (i.e.$a = 75,w = 5$)
Substituting the value of$w$ in $a = 20w - {w^2}$, we get
$a = 20 \times 5 - 25$
$a = 75$is accepted
    Let us consider the next relationship (i.e.$a = 96,w = 8$)
Substituting the value of$w$ in $a = 20w - {w^2}$, we get
$a = 20 \times 8 - 64$
$a = 96$ and it is applicable.
    Let us consider the next relationship (i.e.$a = 100,w = 10$)
Substituting the value of$w$ in $a = 20w - {w^2}$, we get
$a = 20 \times 10 - 100$
$a = 100$ and it is applicable.
    Let us consider the next relationship (i.e.$a = 96,w = 12$)
Substituting the value of$w$ in $a = 20w - {w^2}$, we get
$a = 20 \times 12 - 144$
$a = 96$ and it is applicable.
    Let us consider the next relationship (i.e.$a = 75,w = 15$)
Substituting the value of$w$ in $a = 20w - {w^2}$, we get
$a = 20 \times 15 - 225$
$a = 75$ and it is applicable.
Hence, option C) is correct.
IV) Now, we shall check option D).
     The algebraic expression is$a = 100 - {w^2}$.
    Let us consider the first relationship (i.e.$a = 75,w = 5$)
Substituting the value of$w$ in $a = 100 - {w^2}$, we get
$a = 100 - 25$
$a = 75$is accepted
    Let us consider the next relationship (i.e.$a = 96,w = 8$)
Substituting the value of$w$ in $a = 100 - {w^2}$, we get
$a = 100 - 64$
$a = 36 \ne 96$ and it is not applicable.
Hence, option D) is also incorrect.

So, the correct answer is “Option C”.

Note: Hence, an algebraic equation in option C) matches the above relationship between area and width.
Some formulae to be noted are given below.
  \[Area{\text{ }}of{\text{ }}a{\text{ }}rectangle = length \times width\]
 \[Perimeter{\text{ }}of{\text{ }}a{\text{ }}rectangle = 2 \times length + 2 \times width\]