A ladder leans against the side of a building with its foot 12 ft from the
building. How far from the ground is the top of the ladder and how
long is the ladder if it makes an angle of 70° with the ground? Provide
an Illustration.
h/12 = tan70°
length = √(12^2 + h^2)
To solve this problem, we can use trigonometry. Let's denote the distance from the ground to the top of the ladder as h and the length of the ladder as L.
We are given that the foot of the ladder is 12 ft from the building, which forms the base of a right-angled triangle. The angle between the ladder and the ground is 70°.
Using the trigonometric function tangent (tan), we can relate the angles and sides of a right-angled triangle:
tan(angle) = opposite/adjacent
In this case, the opposite side is h (the height of the ladder) and the adjacent side is 12 ft (the distance of the foot of the ladder from the building).
So, we can write:
tan(70°) = h/12
To find h, we can rearrange the equation:
h = 12 * tan(70°) ≈ 29.48 ft
Therefore, the top of the ladder is approximately 29.48 ft from the ground.
To find the length of the ladder (L), we can use the Pythagorean theorem, which states that in a right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
In this case, one of the sides is 12 ft, and the other side is h (approximately 29.48 ft). Let's calculate L:
L^2 = 12^2 + 29.48^2
L^2 ≈ 144 + 870.86
L^2 ≈ 1014.86
L ≈ √1014.86
L ≈ 31.85 ft
Therefore, the length of the ladder is approximately 31.85 ft.
Here is an illustration of the ladder leaning against the building:
|
|
|\
| \
L | \ h
| \
| \
| \
| \
______|_______\______
12 ft 29.48 ft
To solve this problem, we can use trigonometric ratios, specifically the one involving sine. Here's how to find the height of the ladder and its length:
1. Begin by drawing a diagram to visualize the problem. Draw a vertical line to represent the building, another line from the ground to the top of the ladder, and a line connecting the foot of the ladder with the building.
2. Label the foot of the ladder as point A, the top of the ladder as point B, and the point where the line from the ground intersects the building as point C. Also, label the distance from the ground to the top of the ladder as x and the length of the ladder as L.
3. Since the angle between the ground and the ladder is given as 70 degrees, label this angle as θ.
4. Apply the trigonometric ratio for sine in this scenario, which states that sine(θ) = opposite/hypotenuse. In this case, the opposite side is x, and the hypotenuse is L. Therefore, we can write the equation as sin(70°) = x/L.
5. Rearrange the equation: x = L * sin(70°).
6. We also know that the foot of the ladder is 12 ft from the building, so the distance between point C and point A is 12 ft.
7. Now, we can apply the Pythagorean theorem to find the value of x:
x^2 + 12^2 = L^2
Simplifying:
x^2 = L^2 - 12^2
x = √(L^2 - 144)
8. Substitute this value of x into the equation from step 5: √(L^2 - 144) = L * sin(70°).
9. Square both sides of the equation to eliminate the square root:
L^2 - 144 = L^2 * sin^2(70°)
10. Solve for L:
0 = L^2 * sin^2(70°) - 144
L^2 = 144 / sin^2(70°)
L = √(144 / sin^2(70°))
11. Finally, substitute the value of L back into the equation from step 5 to find x:
x = L * sin(70°)
Now you can calculate the values of x and L using a scientific calculator or by using trigonometric tables.
Please note that the solution provided above assumes that the ground is flat and the ladder is stable.