Question

Dimensions of a box are: height = x inch, length= 24-2x inch, breadth= 9-2x inch. What is the maximum volume of the box?
A) 200 cubic inch
B) 400 cubic inch
C) 100 cubic inch
D) None of the above

Answer 1

Hint:
The equation for the volume of the box is given by the product of its dimensions. But we have to find the maximum volume of the box. So we will differentiate it to x and equate it to zero.

Complete step by step solution:
Formula for finding the volume of the box is\[ = l \times b \times h\].
\[
   \Rightarrow \left( {24 - 2x} \right)\left( {9 - 2x} \right)x \\
   \Rightarrow \left( {24 - 2x} \right)\left( {9x - 2{x^2}} \right) \\
   \Rightarrow \left( {216x - 48{x^2} - 18{x^2} + 4{x^3}} \right) \\
   \Rightarrow 4{x^3} - 66{x^2} + 216x \\
 \]
This is the equation for the volume of the box.
Now to find the maximum volume of the box we have to differentiate the equation with respect to x and then equate it to zero.
\[\dfrac{{dV}}{{dx}} = 12{x^2} - 132x + 216\]
\[ \Rightarrow \dfrac{{dV}}{{dx}} = 12{x^2} - 132x + 216 = 0\]
Dividing the equation by 12.
\[
   \Rightarrow {x^2} - 11x + 18 = 0 \\
   \Rightarrow {x^2} - 9x - 2x + 18 = 0 \\
   \Rightarrow x\left( {x - 9} \right) - 2\left( {x - 9} \right) = 0 \\
   \Rightarrow \left( {x - 9} \right)\left( {x - 2} \right) = 0 \\
 \]
Thus
\[ \Rightarrow x - 9 = 0\] Or \[x - 2 = 0\]
So,
\[ \Rightarrow x = 9\] Or\[x = 2\].
But \[x = 9\] will give us a breadth value of a negative number and this is not acceptable.
So, \[x = 2\] is the value of x.
Now put it in the equation of maximum volume
\[
   \Rightarrow 4{x^3} - 66{x^2} + 216x \\
   \Rightarrow 4 \times {2^3} - 66 \times {2^2} + 216 \times 2 \\
   \Rightarrow 4 \times 8 - 66 \times 4 + 432 \\
   \Rightarrow 32 - 264 + 432 \\
   \Rightarrow 464 - 264 \\
   \Rightarrow 200 \\
 \]
Thus the volume of the box is 200 cubic inch.

Thus option A is correct.

Note:
The important point is to find the maximum volume of the box. This is the key that we have to use differentiation. This way is also used in finding maximum area , maximum deviation etc.