Mathematics Lab Activity-09 Class XII

Mathematics Laboratory Activities 09 on Applications of Derivatives for class XI Non-Medical students with complete observation tables strictly according to the CBSE syllabus.

Chapter - 06 

applications of derivatives

Activity - 09

Objective

To construct an open box of maximum volume from a given rectangular sheet by cutting equal squares from each corner.

Material Required

Chart papers, scissors, cello tape, calculator etc.

Procedure

1. Take a rectangular chart paper of size 20 cm x 10 cm and name it as ABCD.

2. Cut four equal squares each of side x cm from each corner A, B, C and D.

3. Repeat the process by taking the same size of chart papers and different values of x.

4. Make an open box by folding its flaps using cello tape/ adhesive.

Figure 9.1

5. When x = 1, volume of the box (V1) = 18 x 8 x 1 = 144 cm³.

6. When x = 1.5, volume of the box (V₂)  = 17 x 7 x 1.5 = 178.5 cm³.

7. When x = 1.8, volume of the box (V₃) = 16.4 x 6.4 x 1.8 = 188.9 cm³.

8. When x = 2, volume of the box (V₄) = 16 x 6 x 2 = 192 cm³.

9. When x = 2.1, volume of the box (V₅) = 15.8 x 5.8 x 2.1 = 192.4 cm³.

10. When x = 2.2, volume of the box (V₆) = 15.6 x 5.6 x 2.2 = 192.2 cm³.

11. When x = 2.3, volume of the box (V₇) = 15.4 x 5.4 x 2.3 = 191.2 cm³.

12. When x = 2.5, volume of the box (V₈) = 15 x 5 x 2.5 = 187.5 cm³.

13. When x = 3, volume of the box (V₉) = 14 x 4 x 3 = 168 cm³

Observations

1. Volume  V₁ is less than volume V_5.
2. Volume  V₂ is less than volume V_5.
3. Volume  V₃ is less than volume V_5.
4. Volume  V₄ is less than volume V_5.
5. Volume  V₆ is less than volume V_5.
6. Volume  V₇ is less than volume V_5.
7. Volume  V₈ is less than volume V_5.
8. Volume  V₉ is less than volume V_5.
9. ⇒ Volume V₅ of the box is maximum.
10. From the above discussion we see that When x = 2.1cm then volume (V₅) = 192.4 cm³  which is maximum.

Result

Volume of the rectangular box obtained by cutting the square pieces from the corner of a rectangular sheet of dimension 20 x 10  is maximum  when side of the square piece is 2.1 cm.

Applications

This activity is useful in explaining the concepts of maxima/minima of functions. It is also useful in making packages of maximum volume with minimum cost.

VIVA – VOICE

Q1. What are extreme points ?

Ans. These are points from domain of 'f' where we can find maxima or minima.

Q2. What is extreme value of function ?

Ans. The value of function f at an extreme point is called extreme value.

Q3. Are extreme points always critical points ?

Ans. No.

Q4. Local maxima or minima may occur at a critical point. Is it true ?

Ans. Yes

Q5. What is local maximum value of function f(x) = sinx ?

Ans. 1

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