Question

The cost of painting a rectangular metal sheet is the square of its area. If the length and breadth of the rectangle are \[2xy\] and \[3{x^2}y\] respectively, find the cost. Given that the area of a rectangle is the product of its length and breadth.

Answer 1

Hint:
Here, we need to find the cost of the rectangular metal sheet. We will use the rules of exponents and the formula for area of a rectangle to find and simplify the expression for area. Then, we will use the given information and rules of exponents to find the cost of painting the rectangular sheet.
Formula used: The area of a rectangle is given by the formula \[l \times b\], where \[l\] is the length and \[b\] is the breadth.

Complete step by step solution:
We will use the formula for area of a rectangle and the rules of exponents to find the required answer.
The length and breadth of the rectangle are \[2xy\] and \[3{x^2}y\] respectively.
We know that a number raised to the power 1 is equal to itself.
Therefore, we can rewrite \[x\] as \[{x^1}\], and \[y\] as \[{y^1}\].
Thus, we get
Length of the rectangle\[ = 2xy = 2{x^1}{y^1}\]
Breadth of the rectangle\[ = 3{x^2}y = 3{x^2}{y^1}\]
Now, we will find the area of the rectangle.
The area of a rectangle is given by the formula \[l \times b\], where \[l\] is the length and \[b\] is the breadth.
Substituting \[l = 2{x^1}{y^1}\] and \[b = 3{x^2}{y^1}\] in the formula, we get
\[ \Rightarrow \]Area of the rectangle \[ = 2{x^1}{y^1} \times 3{x^2}{y^1}\]
Rewriting the expression, we get
\[ \Rightarrow \]Area of the rectangle \[ = 2 \times 3 \times {x^1} \times {x^2} \times {y^1} \times {y^1}\]
Rewriting the expression using the rule of exponent \[{a^b} \times {a^c} = {a^{b + c}}\], we get
\[ \Rightarrow \]Area of the rectangle \[ = 2 \times 3 \times {x^{1 + 2}} \times {y^{1 + 1}}\]
Multiplying the terms, we get
\[ \Rightarrow \]Area of the rectangle \[ = 6{x^3}{y^2}\]
Thus, the area of the rectangle is \[6{x^3}{y^2}\].
Now, it is given that the cost is square of the area of the rectangle.
Thus, we get
\[ \Rightarrow \]Cost of painting \[ = \] \[{\left( {{\text{Area}}} \right)^2}\]
\[ \Rightarrow \]Cost of painting \[ = \] \[{\left( {6{x^3}{y^2}} \right)^2}\]
Rewriting the expression using the rule of exponent \[{\left( {ab} \right)^c} = {a^c} \times {b^c}\], we get
\[ \Rightarrow \]Cost of painting \[ = \] \[{6^2}{\left( {{x^3}} \right)^2}{\left( {{y^2}} \right)^2}\]
Simplifying the expression using the rule of exponent \[{\left( {{a^b}} \right)^c} = {a^{b \times c}}\], we get
\[ \Rightarrow \]Cost of painting \[ = \] \[{6^2}{x^{3 \times 2}}{y^{2 \times 2}}\]
Multiplying the terms, we get

\[ \Rightarrow \]Cost of painting \[ = \] \[36{x^6}{y^4}\]
Thus, we get the cost of painting the rectangular metal sheet as \[36{x^6}{y^4}\].

Note:
We have used three rules of exponents to simplify and rewrite the solution.
(a) If two or more numbers with the same base and different exponents are multiplied, the product can be written as \[{a^b} \times {a^c} = {a^{b + c}}\].
(b) If two or more numbers with different base and same exponents are multiplied, the product can be written as \[{a^c} \times {b^c} = {\left( {ab} \right)^c}\].
(c) If a number raised to an exponent is again raised to an exponent, the new power is the product of the exponents. This can be written as \[{\left( {{a^b}} \right)^c} = {a^{b \times c}}\].