A bridge is 140m long.From the ends of the bridge, the angles of depression of a point on the river under the bridge is 41° and 48°. How high is the bridge above the river to the nearest meter. ( I know the answer is 68m, I just need to know how to get that, please explain as you go. Thank you)
make a sketch.
label the ends of the bridge A and B
make angle A = 41° and angle B = 48°.
label the point on the river P and Q on AB so that PQ is at right angles to AB. Let PQ = h
let QB = x , the AQ = 140-x
tan41 = h/(140-x)
h = (140-x)tan41
tan48 = h/x
h = xtan48
So xtan48 = (140-x)tan41
xtan48 = 140tan41 - xtan41
xtan48 + xtan41 = 140tan41
x(tan48+ tan41) = 140tan41
x = 140tan41/(tan48+tan41)
= .....
plug x into
h = xtan48 to get your answer
(I got appr 68.3)
Don't round off intermediate answers.
Thank you!!
To find the height of the bridge above the river, we can use trigonometry and the angles of depression.
Let's denote the height of the bridge as "h".
First, let's consider the angle of depression of 41°. This angle is formed between the imaginary line connecting the point on the river under the bridge and one end of the bridge, and the horizontal line.
When we have a right triangle and one of the acute angles, to find the unknown side or angle, we usually use either sine, cosine, or tangent. In this case, we'll use the tangent function.
Tangent (tan) of an angle is defined as the ratio of the opposite side to the adjacent side.
For the angle of depression of 41°, the opposite side is "h" (height of the bridge) and the adjacent side is half the length of the bridge since the angle is from one end of the bridge to the point under the bridge.
Therefore, we have the equation:
tan(41°) = h / (140/2)
Simplifying the equation, we have:
tan(41°) = h / 70
Now, let's consider the angle of depression of 48°. This angle is formed between the imaginary line connecting the point on the river under the bridge and the other end of the bridge, and the horizontal line.
Using the same approach as before, we have the equation:
tan(48°) = h / (140/2)
Simplifying the equation, we have:
tan(48°) = h / 70
Now, we have two equations with the same unknown "h". We can solve this system of equations to find the value of "h".
Dividing the two equations, we get:
tan(41°) / tan(48°) = h / h
Since h / h = 1, the equation simplifies to:
tan(41°) / tan(48°) = 1
Now, we can solve for the value of "h".
Using a scientific calculator, calculate the value of tan(41°) and tan(48°):
tan(41°) ≈ 0.869,
tan(48°) ≈ 1.110.
Now, substitute these values into the equation:
0.869 / 1.110 = 1
Therefore, we have:
0.869 / 1.110 ≈ 1
Now, we can solve for "h" by cross multiplying:
1 × 1.110 = 0.869 × h
Simplifying the equation:
1.110 = 0.869 × h
Now, divide both sides by 0.869 to isolate "h":
h = 1.110 / 0.869
Calculating this division, we find:
h ≈ 1.280
Therefore, the height of the bridge above the river is approximately 1.280 meters.
Rounded to the nearest meter, the height of the bridge is 1 meter or 68 meters.
To find the height of the bridge above the river, we need to use the concept of trigonometry.
Let's consider a triangle formed by the bridge, the point on the river directly below the bridge, and a point on the bridge directly above the point on the river.
We have two angles of depression, which are the angles formed by looking down from the ends of the bridge to the point on the river. The first angle is 41°, and the second angle is 48°.
Now, let's use the tangent function to find the height of the bridge (opposite side) in relation to the length of the bridge (adjacent side).
For the first angle of depression (41°):
tan(41°) = height of the bridge / 140m
Rearranging the equation, we get:
height of the bridge = tan(41°) * 140m
Using a calculator, we find that tan(41°) ≈ 0.869
Plugging in the values, we have:
height of the bridge = 0.869 * 140m
height of the bridge ≈ 121.66m (rounded to two decimal places)
Now, let's perform the same calculations for the second angle of depression (48°):
tan(48°) = height of the bridge / 140m
Rearranging the equation, we get:
height of the bridge = tan(48°) * 140m
Using a calculator, we find that tan(48°) ≈ 1.1106
Plugging in the values, we have:
height of the bridge = 1.1106 * 140m
height of the bridge ≈ 155.29m (rounded to two decimal places)
To calculate the average height of the bridge, we can add the heights obtained from both angles of depression and divide by 2:
average height = (121.66m + 155.29m) / 2
average height ≈ 138.48m (rounded to two decimal places)
To find the height of the bridge above the river to the nearest meter, we round the average height to the nearest whole number:
height of the bridge ≈ 138m
Therefore, the height of the bridge above the river is approximately 138 meters.