Mathematics Lab Activity-10 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 - 10

Objective

To find the time when the area of a rectangle of given dimensions become maximum, if the length is decreasing and breadth is increasing at given rates.

Material Required

Chart paper, paper cutter, scale, pencil, eraser, cardboard.

Procedure

1. Take a rectangle R₁ of dimensions 16 cm x 8 cm.

2. Let the length of the rectangle is decreasing at the rate of 1cm/second and the breadth is increasing at the rate of 2cm/second.

3. Cut other rectangles R₂ , R₃ , R₄, R₅, R₆ , R₇, R₈, R₉, etc. of dimensions 15 cm x 10 cm, 14 cm x 12 cm, 13cm x 14 cm, 12 cm x 16 cm, 11 cm x 18 cm, 10cm x 20 cm, 9 cm x 22 cm, 8cm x 24 cm.

4. Paste all these rectangles on a card board as shown in figure 10.1.

Figure 10.1

5. The length of the rectangle decreasing at the rate of 1cm/s and breadth is increasing at the rate of 2 cm/s.

6. Area of the given rectangle R₁ = 16 x 8 = 128 cm².

7. Area of the given rectangle R₂ = 15 x 10 = 150 cm² (after 1 sec)

8. Area of the given rectangle R₃ = 14 x 12 = 168 cm² (after 2 sec)

9. Area of the given rectangle R₄ = 13 x 14 = 182 cm² (after 3 sec)

10. Area of the given rectangle R₅ = 12 x 16 = 192 cm² (after 4 sec)

11. Area of the given rectangle R₆ = 11 x 18 = 198 cm² (after 5 sec)

12. Area of the given rectangle R₇ = 10 x 20 = 200 cm² (after 6 sec)

13. Area of the given rectangle R₈ = 9 x 22 = 198 cm² (after 7 sec)

Observations

1. Area of the rectangle R₁  = 128 cm².

2. Area of the rectangle R₂ (after 1 second) = 150 cm².

3. Area of the rectangle R₃ (after 2 second) = 168 cm².

4. Area of the rectangle R₄ (after 3 second) = 182 cm².

5. Area of the rectangle R₅ (after 4 second) = 192 cm².

6. Area of the rectangle R₆ (after 5 second) = 198 cm².

7. Area of the rectangle R₇ (after 6 second) = 200 cm².

8. Area of the rectangle R₈ (after 7 second) = 198 cm².

9. Area of the rectangle is maximum after 6 seconds and the Maximum area of the rectangle is  200cm²

Result

Maximum area of the rectangle is 200cm² after 6 seconds.

Applications

This activity is used in explaining the concept of rate of change of quantities

VIVA – VOICE

Q1. Define rate of change of function y = f(x)

Ans. If a quantity y varies with another quantity x such that y = f(x), then    represents the rate of change of y w.r.t x.

Q2. What symbol is used to write rate of change of y w. r. t. t ?

Ans.   .

Q3. What symbol is used to write rate of change of y w. r. t. x ?

Ans.  

Q4. What is chain rule ?

Ans.   .

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