A ladder, 6m long rests against a wall. The foot of the ladder is 2.5m away from the base of the wall. The ladder has a 'safe angle' with the ground between 60(degrees) and 70(degrees). what are the safe limits for the distance of the foot of the ladder to the wall.
If the base of the ladder is at distance b from the wall, and the ladder makes an angle θ with the ground,
b/6 = cosθ
sec 60° = .5
sec 70° = .34
so 6*2 = 3
6*.34 = 2.04
so, the base may safely be between 2 and 3 feet from the wall.
To find the safe limits for the distance of the foot of the ladder to the wall, we need to consider the range of angles between 60 degrees and 70 degrees.
Let's start by drawing a diagram to visualize the problem. We have a ladder that forms a right triangle with the wall and the ground. The length of the ladder is 6m, and the foot of the ladder is 2.5m away from the base of the wall. We can label the height of the triangle (the distance between the top of the ladder and the ground) as 'h' and the distance from the foot of the ladder to the wall as 'x.'
Now, we know that the ladder forms a right triangle, so we can apply trigonometric ratios to solve the problem.
Using the sine ratio: sin(angle) = opposite/hypotenuse
Sine of an angle is equal to the ratio of the length of the side opposite the angle to the length of the hypotenuse.
In this case, we want to find the range of possible values for 'x' that satisfy the given conditions. So, we need to find the range of values for 'x' such that the angle falls within the range of 60 to 70 degrees.
Let's calculate the potential range of values for 'x.'
For the lower angle limit of 60 degrees, sin(60) = h/6.
Rearranging the equation, h = 6 * sin(60) = 6 * 0.866 (rounded to 3 decimal places) = 5.196.
For the upper angle limit of 70 degrees, sin(70) = h/6.
Rearranging the equation, h = 6 * sin(70) = 6 * 0.939 (rounded to 3 decimal places) = 5.634.
Since 'h' represents the height, it cannot be negative. Therefore, we need the range of 'h' to be between 5.196 and 5.634.
Now, we can apply the Pythagorean theorem to relate 'h' and 'x.' According to the theorem, the sum of the squares of the two shorter sides of a right triangle is equal to the square of the hypotenuse.
Using this theorem, we can write an equation:
x^2 + h^2 = 6^2 (since the length of the ladder is 6m)
Substituting the values we found for 'h,' we have:
x^2 + (5.196)^2 = 6^2
x^2 + 26.980 = 36
x^2 = 36 - 26.980
x^2 = 9.02
x = sqrt(9.02) ≈ 3.003
The square root of 9.02 is approximately equal to 3.003. Therefore, the safe limits for the distance of the foot of the ladder to the wall are between approximately 3.003m and 2.5m.