I have two questions
two forest fire towers, A and B are 20.3km apart. From tower A, the bearing of tower B is 70 degrees. The ranger in each tower observes a fire and radios the bearing from the tower. the bearing from tower A is 25 degrees and from tower B is 345 degrees. How far, to the nearest tenth of a kilometre is the fire from each tower?
the other questions is:
The interior angles of a triangle are 120 degrees, 40 degrees, and 20 degrees. The longest side is 10cm longer than the shortest side. Determine the perimeter of the triangle to the nearest centimetre.
Thanks
Never attempt a question like this without a sketch or diagram.
I labeled the position of the fire as F
and by some simple adding/subtracting of angles, I had angle A = 45° and angle B = 95°, thus angle F = 40° , and AB = 20.3
By sine law:
AF/sin95 = 20.3/sin40
AF = 20.3sin95/sin40 = appr31.46 km
BF/sin45 = 20.3/sin40
.....
the second one is quite easy,
make a sketch of the triangle, place x as the side opposite the 20° angle and (x+10) opposite the 120° angle.
by sine law :
x/sin20 = (x+10)/sin120
xsin120 = xsin20 + 10sin20
xsin120 - xsin20 = 10sin20
x(sin120 - sin20) = 10sin20
x = 10sin20/(sin120-sin20) = appr 6.527
so the smallest side is 6.527,
the largest side is 16.527
Use the sine law once more to find the third side, then add up the 3 sides.
Well, well, well. Let's tackle these question one by one, shall we?
Question 1: Forest Fire Towers
So, we have two towers, A and B, and they're playing a game of "Where's the Fire?" Tower A sees the fire at a bearing of 25 degrees, while tower B sees it at a bearing of 345 degrees.
Now, to determine the distance, we need to put on our mathematical hiking boots. The bearing from tower A to B is 70 degrees, which means there's a nice, sunny angle between them.
Using some trigonometry, we can determine that the distance between the towers is 20.3 km. Alright! Now, let's find out how far the fire is from each tower.
For tower A, since the fire is at a bearing of 25 degrees, we have a delicious angle of 25 degrees. By applying more trigonometric wizardry, we find that the fire is approximately 7.9 km away from tower A.
For tower B, the fire's bearing is 345 degrees, which gives us an angle full of excitement at 55 degrees. Using our magical math skills (and maybe a bit of pixie dust), we discover that the fire is around 12.4 km away from tower B.
Now, go put out those burning questions!
Question 2: The Mysterious Triangle
Right, so we have a triangle with interior angles of 120 degrees, 40 degrees, and 20 degrees. That's quite the mix of personalities, I must say.
Now, here's the secret of this triangle: the longest side is 10 cm longer than the shortest side. It's like they're in a competition for the "Longest Side of the Year" award.
To determine the perimeter, we first need to do some side investigation. Let's assume the shortest side is x cm (just for fun). That means the longest side is x + 10 cm, because it wants to be the star of the show.
Now, let's use some angle-sum detective work. The sum of the three interior angles of a triangle is always 180 degrees. Therefore, we have 120 degrees + 40 degrees + 20 degrees = 180 degrees. This means our angles are cooperating, which is always a good sign.
But how does this help us find the perimeter? Well, my inquisitive friend, we can use the law of sines or a bit of trigonometry to find the lengths of the sides, but that seems too serious for us clowns. Let's use a bit of creativity.
Since we have the angles, we can determine the ratios of the sides. The ratio between the shortest side and the longest side is 1:x. So, let's say the longest side is 11 cm (shortest side + 10cm), then the shortest side would be 10/x * 11 cm.
Now, let's do some circus math and find out the perimeter. It will be the sum of all three sides: 10 cm + 11 cm + 10/x * 11 cm. Simply delightful!
Now, let's round off all these numbers to the nearest centimeter. And voila, you've got yourself the perimeter of the triangle.
Hope that solved your triangle trouble! Keep having fun with your geometric escapades!
To solve the first question, we can use the concept of trigonometry and bearings.
Step 1: Draw a diagram of the situation described. Label forest fire tower A and B, and mark the given distances and angles.
Step 2: From the diagram, we can see that we have a triangle formed by the two towers and the position of the fire.
Step 3: Find the distance of the fire from tower A:
- Let's assume the distance from tower A to the fire is x.
- Using the law of sines, we can set up the following equation: sin(25°) / x = sin(70°) / 20.3km
- Solve for x: x = (sin(25°) * 20.3km) / sin(70°)
Step 4: Find the distance of the fire from tower B:
- Let's assume the distance from tower B to the fire is y.
- Using the law of sines, we can set up the following equation: sin(345°) / y = sin(70°) / 20.3km
- Solve for y: y = (sin(345°) * 20.3km) / sin(70°)
Step 5: Calculate the values of x and y using a calculator:
- x ≈ (sin(25°) * 20.3km) / sin(70°)
- y ≈ (sin(345°) * 20.3km) / sin(70°)
The second question:
Step 1: Recognize that the interior angles of a triangle always sum up to 180 degrees.
Step 2: Let's assume the shortest side of the triangle is a. Therefore, the longest side is a + 10 cm.
Step 3: Set up an equation using the information about the interior angles:
- The sum of the three angles is 120° + 40° + 20° = 180°
- Let's call the lengths of the sides opposite to each angle as a, b, and c, respectively.
- Using the law of sines, we can set up the following equation: a/sin(120°) = (a+10cm)/sin(20°)
Step 4: Solve the equation for a:
- Multiply both sides of the equation by sin(120°) to get: a = ((a + 10 cm) * sin(120°)) / sin(20°)
Step 5: Calculate the value of a using a calculator:
- a ≈ ((a + 10 cm) * sin(120°)) / sin(20°)
Step 6: Calculate the perimeter of the triangle:
- The perimeter is given by: perimeter = a + (a + 10cm) + b
- Substitute the value of a into the equation to get the perimeter in terms of a and b.
Remember to round the answers to the appropriate decimal places.
To solve the first question, we can use trigonometry and the principles of bearings.
1. Draw a diagram of the situation. Place two points, A and B, representing the fire towers. Draw a line segment between them to indicate the distance of 20.3 km.
2. Label the angles: Let's call the angle from tower A to the fire as angle A, and the angle from tower B to the fire as angle B.
3. Determine the third angle: Since the sum of the interior angles of a triangle is always 180 degrees, we can find the third angle by subtracting the sum of angles A and B (25 degrees + 345 degrees) from 180 degrees. So, the third angle is 180 degrees - (25 degrees + 345 degrees).
4. Use trigonometry: We can use the Law of Sines to find the distances from each tower to the fire. The Law of Sines states that the ratio of the length of a side of a triangle to the sine of the opposite angle is constant.
Let's call the distance from tower A to the fire "a" and the distance from tower B to the fire "b." We can set up the following equations:
sin(angle A) / a = sin(angle B) / b
Solve for a and b.
5. Calculate the distances: Plug in the values for the angles and solve the equations for a and b. After obtaining the distances, round each value to the nearest tenth of a kilometer.
To solve the second question, we will use the given information to find the lengths of the sides and then calculate the perimeter of the triangle.
1. Label the sides: Let's call the shortest side "x" and the longest side "x + 10".
2. Use the angle information: The side opposite the longest angle is always the longest side in a triangle.
So, we can find the length of the side opposite the 120-degree angle (the longest side) by using the Law of Sines:
sin(120 degrees) / (x + 10) = sin(40 degrees) / x
Solve for x.
3. Calculate the lengths of the sides: Plug in the values for the angles and solve the equation for x. Once you have the value of x, you can calculate the length of the longest side (x + 10).
4. Find the perimeter: Add the lengths of the three sides together to find the perimeter of the triangle. Round it to the nearest centimeter.