Three ships are at points P, Q, and R such that PQ=2km PR=7 km, and
<QPR=142. What is the distance between Q and R, to two decimal places.
To find the distance between points Q and R, we can use the law of cosines. The law of cosines states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides, minus twice the product of the lengths of those two sides multiplied by the cosine of the angle between them.
In this case, we want to find the distance between points Q and R, which can be represented by side QR. We already know that PQ = 2 km and PR = 7 km.
Using the law of cosines, we have the formula:
QR^2 = PQ^2 + PR^2 - 2*PQ*PR*cos(QPR)
Substituting the values we know, we have:
QR^2 = 2^2 + 7^2 - 2*2*7*cos(142°)
Simplifying:
QR^2 = 4 + 49 - 28*cos(142°)
To find the distance, we need to find the cosine of 142°.
cos(142°) ≈ -0.766
Plugging this value back into the equation:
QR^2 ≈ 4 + 49 - 28*(-0.766)
QR^2 ≈ 53 + 21.448
QR^2 ≈ 74.448
Taking the square root to find QR:
QR ≈ √74.448
QR ≈ 8.63 km
Therefore, the distance between points Q and R is approximately 8.63 km, to two decimal places.
To find the distance between points Q and R, we can use the Law of Cosines. The Law of Cosines states that in a triangle, the square of one side (c) is equal to the sum of the squares of the other two sides (a and b), minus twice the product of the lengths of those two sides multiplied by the cosine of the included angle (C).
In this case, we are given the lengths of sides PQ and PR, and the measure of angle QPR. Let's call the unknown side QR.
We have:
PQ = 2 km
PR = 7 km
<QPR = 142 degrees
Using the Law of Cosines, we can write the equation as follows:
QR^2 = PQ^2 + PR^2 - 2 * PQ * PR * cos(<QPR)
Let's calculate it step by step:
1. Convert the angle from degrees to radians:
142 degrees * (π/180) = 2.48 radians (rounded to two decimal places)
2. Substitute the given values into the equation:
QR^2 = (2 km)^2 + (7 km)^2 - 2 * 2 km * 7 km * cos(2.48 radians)
3. Simplify the equation:
QR^2 = 4 km^2 + 49 km^2 - 28 km^2 * cos(2.48 radians)
4. Calculate the value of cosine:
cos(2.48 radians) ≈ -0.795 (rounded to three decimal places)
5. Substitute the value of cosine into the equation:
QR^2 = 4 km^2 + 49 km^2 - 28 km^2 * (-0.795)
6. Simplify the equation further:
QR^2 = 4 km^2 + 49 km^2 + 22.06 km^2
7. Add up the terms:
QR^2 = 75.06 km^2
8. Take the square root of both sides to find the distance QR:
QR = √(75.06 km^2) ≈ 8.66 km
Therefore, the distance between points Q and R is approximately 8.66 km, rounded to two decimal places.
direct application of the cosine law
d^2 = 2^2 + 7^2 - 2(2)(7)cos 142
solve for d