CBSE (Commerce) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Solution for A Rectangular Sheet of Tin 45 Cm by 24 Cm is to Be Made into a Box Without Top, in Cutting off Squares from Each Corners and Folding up the Flaps. ? - CBSE (Commerce) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

A rectangular sheet of tin 45 cm by 24 cm is to be made into a box without top, in cutting off squares from each corners and folding up the flaps. What should be the side of the square to be cut off so that the volume of the box is maximum possible?

Solution

\[\text { Suppose square of side measuring x cm is cut off } . \]

\[\text { Then, the length, breadth and height of the box will be } \left( 45 - x \right), \left( 24 - 2x \right) \text { and } x, respectively . \]

\[ \Rightarrow \text { Volume of the box, V } = \left( 45 - 2x \right)\left( 24 - 2x \right)x\]

\[ \Rightarrow \frac{dV}{dx} = \left( 45 - 2x \right)\left( 24 - 2x \right) - 2x\left( 45 - 2x \right) - 2x\left( 24 - 2x \right)\]

\[\text { For maximum or minimum values of V, we must have }\]

\[\frac{dV}{dx} = 0\]

\[ \Rightarrow \left( 45 - 2x \right)\left( 24 - 2x \right) - 2x\left( 45 - 2x \right) - 2x\left( 24 - 2x \right) = 0\]

\[ \Rightarrow 4 x^2 + 1080 - 138x - 48x + 4 x^2 + 4 x^2 - 90x = 0\]

\[ \Rightarrow 12 x^2 - 276x + 1080 = 0\]

\[ \Rightarrow x^2 - 23x + 90 = 0\]

\[ \Rightarrow x^2 - 18x - 5x + 90 = 0\]

\[ \Rightarrow x\left( x - 18 \right) - 5\left( x - 18 \right) = 0\]

\[ \Rightarrow x - 18 = 0 or x - 5 = 0\]

\[ \Rightarrow x = 18 or x = 5\]

\[\text { Now,} \]

\[\frac{d^2 V}{d x^2} = 24x - 276\]

\[ \frac{d^2 V}{d x^2}_{x = 5} = 120 - 276 = - 156 < 0 \]

\[ \frac{d^2 V}{d x^2}_{x = 18} = 432 - 276 = 156 > 0\]

\[\text { Thus, volume of the box is maximum when x } = 5 cm . \]

\[\text { Hence, the side of the square to be cut off measures } 5 cm .\]

  Is there an error in this question or solution?

APPEARS IN

Video TutorialsVIEW ALL [1]

Solution for question: A Rectangular Sheet of Tin 45 Cm by 24 Cm is to Be Made into a Box Without Top, in Cutting off Squares from Each Corners and Folding up the Flaps. ? concept: Graph of Maxima and Minima. For the courses CBSE (Commerce), CBSE (Arts), PUC Karnataka Science, CBSE (Science)
S
View in app×