A tightrope walker attaches a cable to the roofs of two adjacent buildings. The cable is 23.5 m long. The angle of inclination of the cable is 13 degrees. The shorter building is 12.0 m high. What is the height of the taller building to the nearest tenth of a metre?
I tried drawing it but it didn't work. I don't which sides to label please help :(
the diagram looks rather routine
draw the smaller building and mark is as 12 m high
draw the other building a bit taller .
draw a horizontal from the smaller to the taller, label the difference in height as x
draw in your cable,
Don't you have a right-angled triangle with a base angle of 13°, opposite as x and hypotenuse as 23.5 M ?
so sin 13° = x/23.5
x = 23.5sin13
= 5.2863..
= 5 m to the nearest metre
So the height of the taller building is 12 + 5 or 17 metres.
To solve this problem, let's start by labeling the sides of the triangle formed by the cable and the buildings.
Let's call the height of the taller building 'x'. The shorter building is given as 12.0 m.
Now, let's label the other sides of the triangle. The cable is given as 23.5 m. We'll call the horizontal distance between the two buildings 'd'.
To find the height of the taller building, we need to use trigonometry. Since we have the angle and the opposite side, we can use the tangent function.
The tangent of an angle is the ratio of the opposite side to the adjacent side. In this case, the opposite side is the height of the taller building (x), and the adjacent side is the horizontal distance between the buildings (d).
The formula for the tangent function is:
tan(theta) = opposite / adjacent
In this case, we have:
tan(13 degrees) = x / d
Now, to find the height of the taller building (x), we need to solve for x.
Rearranging the equation, we have:
x = tan(13 degrees) * d
To find the value of d, we can use the Pythagorean theorem:
d^2 = 23.5^2 - 12^2
Simplifying this equation, we have:
d^2 = 551.75 - 144
d^2 = 407.75
Now, we can find the square root of both sides to find the value of d:
d ≈ √407.75
Using a calculator, we find that d ≈ 20.1943
Now, substituting the value of d back into our equation for x, we have:
x = tan(13 degrees) * 20.1943
Calculating this using a calculator, we find that x ≈ 4.8
Therefore, the height of the taller building to the nearest tenth of a meter is approximately 4.8 meters.
To solve this problem, you'll need to use trigonometry, specifically the concept of sine and the sine ratio. The angle of inclination and the side opposite to it will be of particular importance.
Here's how you can label the sides of the triangle formed by the cable and the buildings:
- Let's call the shorter building A and the taller building B.
- The side opposite the angle of inclination of 13 degrees is the height of building A, which we'll label as "a" (12.0 m).
- The side adjacent to the angle of inclination is the horizontal distance between the bases of the two buildings. Let's label it as "b."
- The hypotenuse of the triangle is the length of the cable, which is 23.5 m. We'll label it as "c."
The trigonometric ratio that relates the angle of inclination with the opposite and hypotenuse is the sine ratio. In this case, we have:
sin(13 degrees) = a / c
Now, we need to solve for the unknown side, which is the height of building B. Let's label it as "x."
sin(13 degrees) = x / c
Since we know the value of c (23.5 m) and sin(13 degrees), we can rearrange the equation to solve for x:
x = sin(13 degrees) * c
Now, plug in the values:
x = sin(13 degrees) * 23.5
Using a calculator, evaluate the sine of 13 degrees and multiply it by 23.5 to find the height of the taller building, rounded to the nearest tenth of a meter.