Gradients

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.

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.

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.

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

\ \ 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 °

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 ° \

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. \

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)).

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 : 20 000 cm 000 40 = m 400 = 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 : 20 000 cm 000 40 = m 400 = distance horizontal vertical = road of gradient So, AC m 400 100 = 4 1 : =

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 : 10 000 (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 : 10 000 cm 000 20 = m 200 =

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 : 10 000 (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 =

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 : 10 000 (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 14.04 . ° \