Sally and Kate stand some distance away from a building which is 10m high. Sally measures the angle of elevation of the top of the building to be 35(degree). From Kate's position, the angle of elevation of the top of the building is 50(degree). If the angle SOK between Sally, Kate, and bottom of the building is 60(degree). Find
A: the distance between Sally and Kate, correct to the nearest meter?
B: the angle SBK?
I tried finding the hypotenuse and solve but I get to a dead end. Any help is appreciated
Thanks Alot.
To solve this problem, we can start by breaking down the given information and drawing a diagram.
Let's label the points as follows:
- S: Sally's position
- K: Kate's position
- O: Bottom of the building
- T: Top of the building
We are given the following angles:
- Angle SOK = 60 degrees
- Angle SOA = 35 degrees
- Angle KOB = 50 degrees
Now, let's solve for the unknowns step by step:
Step 1: Find the distance between Sally and the building (SA).
To find SA, we can use the tangent function. The tangent of angle SOA is equal to the opposite side (OT) divided by the adjacent side (SA).
Tan(35) = OT / SA
Since OT is the height of the building (10m), we can rearrange the equation to solve for SA:
SA = OT / Tan(35)
SA = 10 / Tan(35)
Using a calculator, we find that SA ≈ 14.28m (rounded to two decimal places).
Step 2: Find the distance between Kate and the building (KA).
We know that the distance between Sally and Kate is the hypotenuse of the right-angled triangle SKA. To find KA, we can use the Law of Cosines.
KA^2 = SA^2 + SK^2 - 2 * SA * SK * Cos(60)
KA^2 = (14.28)^2 + SK^2 - 2 * 14.28 * SK * (1/2)
Since we are given that the angle SOK is 60 degrees, we can simplify the equation further:
KA^2 = (14.28)^2 + SK^2 - 14.28 * SK
KA^2 = (14.28)^2 + SK^2 - 7.14 * SK
Step 3: Find the value of the angle SBK.
To find the angle SBK, we can use the Law of Sines.
Sin(SBK) / SK = Sin(KSB) / KA
Sin(SBK) / SK = Sin(50) / KA
Let's rearrange this equation to solve for Sin(SBK):
Sin(SBK) = (Sin(50) / KA) * SK
Using the values we've already found, we can substitute them into the equation:
Sin(SBK) = (Sin(50) / √(KA^2 - SK^2 + 7.14SK))
Using a calculator, we find that Sin(SBK) ≈ Sin(50) / √(KA^2 - SK^2 + 7.14SK)
Finally, we can find the angle SBK using the inverse Sin function:
SBK = Sin^(-1)(Sin(SBK))
Using a calculator, we find that SBK ≈ Sin^(-1)(Sin(SBK))
By following these steps, you should be able to find the values for both A and B. Remember to round your answers as specified in the question.
Apparently
O = base of building
B = top of building
If so, then
10/SO=tan35°
SO=14.28
10/KO=tan50°
KO=8.39
SK^2=SO^2+KO^2-2*SO*KO*cos60°=14.28^2+8.39^2-2*14.28*8.39*0.5=154.50
SK=12.43
Now use the same steps to find SB, KB, and angle SBK.