thebottom of the ladder is 3 feet from the building. the ladder makes an angle of 75 degree with the ground. how high up on the building does the ladder reach? how long is the ladder?
length is 5.2m
height is 2.6 ft
To find out how high up on the building the ladder reaches, we need to use trigonometry.
Step 1: Let's label some important values. The bottom of the ladder is 3 feet from the building, so we have the adjacent side, which we'll label as 'adj' = 3 ft. The angle between the ground and the ladder is 75 degrees.
Step 2: We can use the trigonometric function cosine (cos) to find the length of the ladder. The cosine of an angle is the ratio of the adjacent side to the hypotenuse. In this case, the hypotenuse is the length of the ladder, which we'll label as 'hyp'.
cos(75°) = adj / hyp
Step 3: Rearrange the equation to solve for the hypotenuse:
hyp = adj / cos(75°)
Step 4: Plug in the values:
hyp = 3 ft / cos(75°)
Step 5: Use a calculator (or consult a trigonometric table) to evaluate cos(75°):
cos(75°) ≈ 0.2588
Step 6: Substitute the value for cos(75°) back into the equation:
hyp = 3 ft / 0.2588
Step 7: Calculate the value for the hypotenuse:
hyp ≈ 3 ft / 0.2588 ≈ 11.587 ft
So, the length of the ladder is approximately 11.587 feet.
Step 8: To find out how high up on the building the ladder reaches, we can use the sine (sin) function.
sin(75°) = opp / hyp
We know that the opposite side is the height on the building where the ladder reaches, which we'll label as 'opp'.
Step 9: Rearrange the equation to solve for opp:
opp = sin(75°) * hyp
Step 10: Plug in the values:
opp = sin(75°) * 11.587 ft
Step 11: Use a calculator (or consult a trigonometric table) to evaluate sin(75°):
sin(75°) ≈ 0.9659
Step 12: Substitute the value for sin(75°) back into the equation:
opp = 0.9659 * 11.587 ft
Step 13: Calculate the value for opp:
opp ≈ 0.9659 * 11.587 ft ≈ 11.190 ft
So, the ladder reaches a height of approximately 11.190 feet up on the building.
To determine how high up on the building the ladder reaches and how long the ladder is, we can use trigonometric functions based on the given angle and the distance between the bottom of the ladder and the building.
Let's solve it step by step:
Step 1: Visualize the problem
Imagine a right-angled triangle where the ladder is the hypotenuse (the longest side), the height on the building is one of the legs, and the horizontal distance between the bottom of the ladder and the building is the other leg.
Step 2: Identify the known values
- The angle between the ladder and the ground is given as 75 degrees.
- The distance between the bottom of the ladder and the building is given as 3 feet.
Step 3: Determine which trigonometric function to use
Since we are given an angle and a side length, we can use the trigonometric function called "sine" to find the height of the ladder and the "cosine" function to find the length of the ladder.
Step 4: Calculate the height of the ladder
To find the height of the ladder on the building, we can use the sine function:
sin(θ) = opposite/hypotenuse
where θ is the given angle (75 degrees), and the opposite side is the height of the ladder.
Rearranging the equation to solve for the height:
height = sin(θ) * hypotenuse
Step 5: Substitute the values and calculate
height = sin(75°) * hypotenuse
To find the value of sin(75°), we can use a calculator. The result is approximately 0.9659.
height = 0.9659 * hypotenuse
Step 6: Calculate the length of the ladder
To find the length of the ladder, we can use the cosine function:
cos(θ) = adjacent/hypotenuse
where θ is the given angle (75 degrees), and the adjacent side is the distance between the bottom of the ladder and the building.
Rearranging the equation to solve for the hypotenuse:
hypotenuse = adjacent / cos(θ)
Step 7: Substitute the values and calculate
hypotenuse = 3 feet / cos(75°)
To find the value of cos(75°), we can use a calculator. The result is approximately 0.2588.
hypotenuse = 3 feet / 0.2588
Step 8: Calculate the height and length
Using the calculated values from steps 5 and 7:
height = 0.9659 * hypotenuse
length = hypotenuse
Substituting the values:
height = 0.9659 * (3 feet / 0.2588)
length = 3 feet / 0.2588
Simplifying the calculations:
height ≈ 11.13 feet
length ≈ 11.57 feet
Therefore, the ladder reaches approximately 11.13 feet up on the building, and the length of the ladder is approximately 11.57 feet.
This is a trigonometry problem.
The length of the adjacent side to 75 degree angle is 3 feet, the hypotenuse is the length of the ladder, and the height that the ladder reaches up the building is the length of the opposite side to the 75 degree angle.
using trig:
tan 75 = opp/3
solve for the opposite side which is the height the ladder reaches along the building
cos 75 = 3/hyp
where hyp is the length of the ladder
solve for hyp