A fireman is standing 25 m due west of the burning building. If the angle of elevation of the ladder is 51°, how long is his ladder?

sinTheta=25/L

solve for L

26

34 meters

40

To find the length of the ladder, we can use trigonometry. Let's take a look at the situation.

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.