In which year was there the greatest difference between the sales as compared to its previous year

Class VIII Math
NCERT Solution for Introduction to Graphs

Exercise � 15.1

1. The following graph shows the temperature of a patient in a hospital, recorded every hour.

(i) What was the patient's temperature at 1 p.m.?

(ii) When was the patient�s temperature 38.5°C?

(iii) The patient's temperature was the same times during the period given. What were two times?

(iv)What was the temperature at 1.30 p.m.? How did you arrive at your answer?

(v) During which periods did the patient�s temperature showed an upwart trend?

Ans. In the graph, we find that the time (in hours) are represented on the x-axis and the temperatee (in °C) are represented on the y-axis. The temperature are any time can be read from the graph exactly in the same way as we read the coordinates of a point. From the graph, we observe that:

(i) The temperature of the patient a: 1 pm 36.5°C.

(ii) The temperature of the patient was 38.5°C at 12 noon.

(iii) The temperature of the patient at 1.30 pm and 2 p.m.

(iv) The temperature of the patient at 1.30 pm was 36.5°C. The point between 1 pm and 2 pm on the x-axis is equidistant from the two points showing 1 pm and 2 pm, so it will represent 1.30 pm. Similarly, the point on the y-axis, between 36°C and 37°C will represent 36.5°C

(v) During the periods 9 am to 10 am, 10 am to 11 am and 2 pm to 3 pm the patients temperature showed an upwards trend.

2. The following lines graph shows the yearly sales 1 figures for a manufacturing company.

(i) What-were the sales in (a) 2002 (b) 2006?

(ii) What were the sales in (a) 2003 (b) 2005?

(iii) Compute the difference between the sales in 2002 and 2006.

(iv) In which year was there the greatest difference between the sales as comapred to its previews year?

Ans. In the graph, we find that the years are presented on the x-axis and the sales (in Rs. crores) on the y-axis. The sales at any time (year) can be read from the graph exactly in the same way as we read the coordinates of a point. From the graph, we observe that:

(i) (a) The sales in the year 2002 is Rs. 4 crore.

(b) The sales in the year 2006 is Rs. 8 crore.

(ii) (a) The sales in the year 2003 is Rs. 7 crore.

(b) The sales in the year 2005 is Rs. 10 crore.

(iii) The difference between the sales in 2002 and 2006 = Rs. 8 crore - Rs. 4 crore = Rs. 4 crore.

(iv) The year in which there was the greatest difference between the sales compared to its previous year is the year 2005.

3. For an experiment in Botany, two different plants, plant A and plant B were grown under similar laboratory conditions. Their heights were measured at the end of each week for 3 weeks. The results are shown by the following graph:

(a) How high was Plant A after (i) 2 weeks (ii) 3 weeks?

(b) How high was Plant B after (i) 2 weeks (ii) 3 weeks?

(d) How much did plant B grow from the end of the 2nd week to the end of the 3rd week?

(e) During which week did Plant A grow maximum ?

(f) During which week did Plant B grow least?

(g) Were the two plants of the same height during any week shown here? Specify.

Ans. In the graph, we find that the weeks are represented on the x-axis and the heights (in cm) of two plants A and B on the y-axis. The heights on any week of plants A and B can be read from the graph exactly in the same way as we read the coordinates of a point. From the graph, we observe that:

(a) The height of the plant A after

(i) 2 weeks was 7 cm

(ii) 3 weeks was 9 cm

(b) The height of the plant B after

(i) 2 weeks was 7 cm

(ii) 3 weeks was 10 cm

(d) The plant B grows 3 cm from the end of the 2nd week to the end of the 3rd week.

(e) The plant A grows most during second week.

(f) The plant B grows most during first week.

(g) At the end of the 2nd week the heights of the two plants were the same.

4. The following graph shows the temperature forecase and the actual temperature for each day of a week.

(a) On which days was the forecast temperature the same as the actual temperature?

(b) What was the maximum forecast temperature during the week?

(d) On which day did the actual temperature differ the most from the forecast temperature?

Ans. In the graph, we find that the days are reresented on the x-axis and the temperature forecast/actual (°C) on the y-axis. The temperature on any day can be read from the graph exactly in the same way as we read the coordinates of a point. From the graph, we observe that:

(a) The days on which the forecast temperature was the same as the actual temperature are Thuesday, Friday and Sunday.

(b) The maximum forecast temperature during the week was 35°C.

(d) The actual temperature differed the most from the forest temperature on Thursday.

5. Use the tables below to draw linear graphs,.

(a) The number of days a hill side city received snow in different years.

(b) Population (in thousands) of men and women in a village in different years.

Ans. (a) In order to draw the required graph, we represent years on the x-axis and the days on the yaxis. We first plot the ordered pairs (2003, 8), (2004, 10), (2005, 5) and (2006 , 12) as points and then join them by line segments as shown below:

(b) In order to draw the required graph, we represent years on the x-axis and the population (in thousands) on the y-axis. The dotted line shows the population (in thousands) of men and the solid line shows the population (in thousands) of women.

We first plot the ordered pairs (2003,12), (2004,12.5), (2005,13), (2006,13.2) and (2007,13.5) and then join them by the dotted line as shown, to get the graph representing the number of men. Further, we plot (2003, 11.2), (2004, 11.9), (2005, 13), (2006,13.6) and (2007,12.8) and then join them by line segments as shown to get the graph representing the graph of number of women. Thus, the required graph is obtained.

6. A courier-person cycles from a town to a neighbouring suburban area to deliver a parcel to a merchant. His distance from the town at different times is shown by the following graph:

(a) What is the scale taken for the time axis

(b) How much time did the person take for the travel?

(d) Did the person stop on his way ? Explain.

(e) During which period did he ride fastest?

Ans. In the graph, we find that the time (in hours) is represented on the x-axis and the distance (in km) is represented on the y-axis. The distance at any time can be read from the graph exactly in the same way as we read the coordinates of a point. From the graph, we observe that:

(a) The scale taken for the time axis is:

4 units = 1 hour.

(d) Yes; this is indicated by the horizontal part of the graph (10 a.m. � 10.30 a.m.)

(e) Between 8 a.m. and 9 a.m. he ride faster.

7. Can there be a time temperature graph as follows? Justify your answer.

Ans. (i) This represents a time-temperature graph because it represents a smooth rise in temperature and is represented by a line graph.

(ii) This represents a time-temeprature graph because it represent a smooth fall in temeprature and is represented by a line graph.

(iii) This does not represent a time-temperature because it shows different temperatures at the same time.

(iv) This represents a time-temperature graph because it shows a constant temperature at different times and is a line graph.

1. Plot the following points on a graph sheet. Verify if they lie on a line

(a) A(4, 0), B(4, 2), C(4, 6), D(4, 2.5)

(b) P(1, 1), Q(2, 2), R(3, 3), S(4, 4)

Ans. (a) Draw the x-axis and y-axls. Plot the points A (4,0), B (4. 2), C (4, 6) and D (4, 2.5) as shown.

Clearly, these points lie on a line ABDC.

2. Draw the line passing through (2,3) and (3,2). Find the coordinates of the points at which, this line meets the x-axis and y-axis.

Ans. Try yourself

3. Write the coordinates of the vertices of each of the figures given on next page.

Ans. Clearly, from the graph the coordinates of points are:

O (0, 0), A (2, 0), B (2, 3) and C (0, 3);

P (4, 3), Q (6, 1), R (6, 5) and S (4, 7);

L (7, 7), M (10, 8) and K (10, 5).

4. State whether True or False. Correct that are false.

(i) A point whose x-coordinate is zero and y-coordinate is non-zero will lie on the y-axis.

(ii) A point whose y-coordinate is zero and x-coordinate Is 5 will lie on y-axis.

(iii) The coordinates of the origin are (0,0).

Ans. (i) True.

(ii) False. Correct statement is: A point whose y-coordinate is zero and x-coordinate is 5 will lie on X-axis

(iii) True

1. Draw the graphs for the following tables of values with suitable scales on the axes.

(i) How much distance did the car cover during the period 7.30 a.m. to 8 a.m. ?

(ii) What was the time when the car had covered a distance of 100 km since it's start?

(i) Does the graph pass through the origin?

(ii) Use the graph to find the interest on Rs 2500 for a year.

(iii) To get an interest of Rs 280 per year, how much money should be deposited?

Ans. (a) In order to represent the given data graphicaly, we represent 'Number of apples' on the x-axis and 'Cost of apples (in Rs)� on the y-axis. The ordered pairs (1, 5)) (2, 10) (3, 15) (4, 20) and (5, 25) are plotted as points and are joined by the line segments as shown in the figure. This gives the required graph.

Ans. (i) In order to represent the given data graphically, represent the side of square (in cm) on the x axis and the perimeter (in cm) on the y-axis.

Scale: 1 unit = 1 cm on x-axis and y-axis.

Plot the points (2, 8), (3,12), (3.5, 14), (5, 20) and (6, 24), Joip these points to obtain the required graph.

It is a linear graph.

Plot points (2, 4), (3, 9), (4, 16), (5, 25), (6,36) It is not a linear graph.