sinTheta=25/L
solve for L
solve for L
The fireman is standing at a point 25 m due west of the burning building. Let's label this point as A. The ladder is inclined at an angle of 51° with the horizontal line. We can label the length of the ladder as 'l', and the point where the ladder touches the building as B.
Now, let's consider a right-angled triangle formed by the fireman, the ladder, and the ground. In this triangle, the angle opposite to the base (25 m) is 90°, the angle opposite to the ladder (l) is 51°, and the angle opposite to the height of the building (AB) is 39° (since the sum of angles in a triangle is 180°).
To find the length of the ladder, we can use the trigonometric function called sine.
sin(angle) = opposite / hypotenuse
In this case, the sine of the angle of elevation (51°) is equal to the side opposite to the angle (AB) divided by the hypotenuse (ladder length, l).
sin(51°) = AB / l
We want to find the value of l, so let's rearrange the equation:
l = AB / sin(51°)
To find the length of AB, we can use the trigonometric function called cosine.
cos(angle) = adjacent / hypotenuse
In this case, the cosine of the angle (39°) is equal to the side adjacent to the angle (AB) divided by the hypotenuse (ladder length, l).
cos(39°) = AB / l
Let's rearrange this equation to find the value of AB:
AB = cos(39°) * l
Now, we can substitute the value of AB in the equation for l:
l = (cos(39°) * l) / sin(51°)
Let's isolate the value of l:
l * sin(51°) = cos(39°) * l
Dividing both sides of the equation by l:
sin(51°) = cos(39°)
Using the trigonometric identities of sine and cosine, we can rewrite the equation as:
sin(51°) = sin(90° - 39°)
Since the sine function is equal for two angles that are complementary (90° - 39° = 51°), we can conclude that:
51° = 90° - 39°
Now we know that this is true. So the equation is:
l = 25 m / sin(51°)
Using a calculator, we can find:
l ≈ 31.804 m
Therefore, the length of the fireman's ladder is approximately 31.804 meters.