1. Gradients

2. Gradient of a Road The gradient shows the steepness of the roads.
On some inclined roads, we may find road signs with ratios (e.g. 1 : 6) showing the gradients of the roads.

3. What is the meaning of gradient = 1 : 6?
1 unit Q P
Gradient of road PQ 6 1 =
The ratio 1 : 6 means that when the change in vertical distance is 1 unit,
the change in horizontal distance is 6 units.

4. If the inclined road makes an angle with the horizontal, then
1 unit Q P
Gradient of road PQ 6 1 =
If the inclined road makes an angle with the horizontal, then
We call the inclination of the road.

5. In general, we have:
distance
horizontal
vertical
Gradient of a road =
If an inclined road makes an angle θ with the horizontal, then

6. \ \ Let’s study the following example.
The photo shows a ramp with gradient 1 : 12.
Let be the inclination of the ramp. q 12 1 =
tan q 8 . 4
(cor. to the nearest 0.1 ) = \ q \ .
4.8
The inclination of the ramp is

7. Follow-up question
Joe is walking up a steep road of gradient 1 : 10.
(a) Find the inclination of the road, correct to the nearest 0.01°.
(b) If Joe travels a horizontal distance of 20 m, what is the vertical distance he has travelled?
Solution
(a)
Let be the inclination of the road. q 10 1
 Gradient =
tan = q \ .
5.71 is
road
the of
inclination
The \

8. Follow-up question (cont’d)
Joe is walking up a steep road of gradient 1 : 10.
(a) Find the inclination of the road, correct to the nearest 0.01°.
(b) If Joe travels a horizontal distance of 20 m, what is the vertical distance he has travelled?
Solution
(b)
Vertical distance = 2 m \
The required vertical distance is 2 m. \

9. Finding Gradient from a Map
Figure (1)
Figure (2)
Figure (1) shows the contour map of an island. The curves marked with 100 m, 200 m and 300 m are called contour lines.
Any two points along the same contour line (e.g. A and B) have the same vertical distance from sea level (see Figure (2)).

10. Let’s try to find the gradient of road AC.
In the figure, AC is a straight inclined road. of
distance
Vertical AC m
100)
(200 - = m
100 =
If AC is measured to be 2 cm on the map, then
horizontal distance of AC cm
000 20 2  =
 Scale = 1 : cm
000 40 = m
400 =
Let’s try to find the gradient of road AC.

11. In the figure, AC is a straight inclined road.of
distance
Vertical AC m
100)
(200 - = m
100 =
If AC is measured to be 2 cm on the map, then
horizontal distance of AC cm
000 20 2  =
 Scale = 1 : cm
000 40 = m
400 =
distance
horizontal
vertical =
road of
gradient
So, AC m
400
100 = 4 1 : =

12. Follow-up question
A student measures AB to be 2 cm on the contour map as shown in the figure.
150 m
100 m
50 m A B
scale 1 :
(a) Find the gradient of road AB in the form 1 : n.
(b) Find the inclination of road AB, correct to the nearest 0.01°.
Solution
(a) m
50)
(100 of
distance
Vertical - = AB m 50 = cm
000 10 2 of
distance
Horizontal  = AB
 Scale = 1 : cm
000 20 = m
200 =

13. Follow-up question (cont’d)
A student measures AB to be 2 cm on the contour map as shown in the figure.
150 m
100 m
50 m A B
scale 1 :
(a) Find the gradient of road AB in the form 1 : n.
(b) Find the inclination of road AB, correct to the nearest 0.01°.
Solution
(a) ∴
distance
horizontal
vertical
road of
Gradient = AB m
200 50 = 4 : 1 =

14. Follow-up question (cont’d)
A student measures AB to be 2 cm on the contour map as shown in the figure.
150 m
100 m
50 m A B
scale 1 :
(a) Find the gradient of road AB in the form 1 : n.
(b) Find the inclination of road AB, correct to the nearest 0.01°.
Solution
(b)
Let be the inclination of road AB. q =
tan q
gradient of AB 4 1 = \ )
0.01
nearest
the to
(cor.
14.04 = q
The inclination of road AB is \