When one end of a ladder,LM,is placed against a vertical wall at a point 5m above the ground,the ladder makes an angle of 37° with the horizontal ground.
1. Represent the information in a diagram
2. Calculate, correct to 3 significant figure,the length of the ladder.
3. If the foot of the ladder is pushed torwards the wall by 2m , calculate,correct to the nearest degree,the angle which the ladder now makes with the ground.
Diagram
1. To represent the information in a diagram, you can draw a right-angled triangle.
Let L represent the foot of the ladder, M represent the top of the ladder, and N represent the base of the ladder where it touches the ground. The vertical wall is represented by a vertical line and the horizontal ground is represented by a horizontal line.
Draw a line from M to N to represent the ladder. Then draw a line from M to the vertical wall. Finally, draw a line from N to M to complete the triangle.
2. To calculate the length of the ladder, you can use trigonometry. In the right-angled triangle, the angle between the ladder and the horizontal ground is 37 degrees and the side opposite this angle is 5m (the height of the wall).
Using the sine ratio:
sin(37°) = opposite/hypotenuse = 5m/hypotenuse
Rearranging the equation to solve for the hypotenuse (the length of the ladder):
hypotenuse = opposite/sin(37°) = 5m/sin(37°)
Calculating this value, using a calculator (sine of 37 degrees ≈ 0.6018):
hypotenuse ≈ 5m/0.6018 ≈ 8.309m
So, the length of the ladder is approximately 8.309 meters (to 3 significant figures).
3. If the foot of the ladder is pushed toward the wall by 2m, we can redraw the triangle with the base (NM) reduced to 2m.
Now we need to find the new angle the ladder makes with the ground. Let's call this angle θ.
Using the tangent ratio:
tan(θ) = opposite/adjacent = 5m/(hypotenuse - 2m)
We already know the value of the hypotenuse (8.309m) from the previous calculation. Substituting these values into the equation, we get:
tan(θ) = 5m/(8.309m - 2m) = 5m/6.309m
Using a calculator:
θ ≈ arctan(5/6.309) ≈ 39.3°
Therefore, the angle that the ladder now makes with the ground is approximately 39.3 degrees (to the nearest degree).
sin 37 = 5/L
so
L = 5/sin 37
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d = original distance along floor from wall
cos 37 = d/L
solve for d
new distance from wall = (d-2)
cos Angle = (d-2)/L
solve for angle