Solve the mathematics question:A town Q is on a bearing of 210 from P; R is another town on a bearing of 150 from P is east of Q.The distance between R and P is 10 km.Find the distance between R and Q.
Solve The question.
I made a sketch and in my triangle PQR , I have angle P = 60°, and PR = 10
That is not enough information, you need 3 bits of independent info.
Perhaps another angle or a side ?
since R is directly east of P, triangle PQR is equilateral, because the angle between PQ and PR is 60°, bisected by the N-S line. So RQ = 10
I think the wording is rather ambiguous, oobleck interpreted it as P is directly east of Q.
I read it that P is to the right of Q, in a generally eastern direction.
Wisdom
Oh, math time! Don't worry, I promise no "bearing" or "east" jokes. Let's solve this!
If R is 10 km east of P and Q is on a bearing of 210 from P, we need to find the distance between R and Q.
Now, to make things simpler, let's call the distance between R and Q "x."
Since R is east of P and Q is on a bearing of 210, we can draw a right-angled triangle with the distance between R and P as the base, Q as the vertical side, and the distance between P and Q as the hypotenuse.
Now, using a bit of trigonometry magic (and exiling any terrible puns from the equation), we can find x using the formula:
sin(θ) = x / 10 km
where θ is the angle between the hypotenuse and the base. In this case, θ would be 210 - 150 = 60 degrees.
Plugging in the values, we get:
sin(60 degrees) = x / 10 km
Simplifying, we find that x = 10 km * sin(60 degrees).
So, the distance between R and Q comes out to be approximately 8.66 km.
To solve this question, we need to visualize the situation and use trigonometry concepts.
First, let's draw a diagram to represent the given information:
```
P
/|
/ |
210/ |10 km
/ |
/ |
Q-----R
150
```
From the diagram, we see that P is the common point for both R and Q.
We are given that the distance between R and P is 10 km. Now let's focus on the bearings mentioned in the question.
A bearing is an angle measured clockwise from the north direction. In this case, town Q is on a bearing of 210° from P, and town R is on a bearing of 150° from P.
To find the distance between R and Q, we can use the concept of angles in a triangle. Since triangle PRQ is a right triangle (with the right angle at P), we can use trigonometry to find the missing side.
We will use the sine function to find the length of side QR. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In this case, the opposite side is QR, and the hypotenuse is PR.
Let's define our variables:
- QR: distance between town Q and town R (what we want to find)
- PR: distance between town P and town R (given as 10 km)
Using the sine function, we can write the equation as:
sine of angle Q = opposite / hypotenuse
sine of 30° (since 180° - 150° = 30°) = QR / 10 km
To find the length of side QR, we rearrange the equation:
QR = sine of 30° * 10 km
Now, let's use a calculator to find the sine of 30°:
sin(30°) ≈ 0.5
Substituting this back into our equation, we find:
QR = 0.5 * 10 km
Therefore, the distance between town R and Q is:
QR = 5 km.