A ground observer sights a weather balloon to the east at an angle of elevation of 15º. A second observer 3 miles to the east of the first also sights the balloon to the east at an angle of elevation of 24º. How high is the balloon?
make a sketch, labeling the balloon P and the point directly below it on the ground as Q
(We have to find PQ)
Label the first observer as A and the second as B
AB = 3
angle PAB = 15° , angle PBQ = 24°
In triangle PAB,
angle A = 15, angle PBA = 156° , so angle APB = 9°
by the sine law:
AP/sin15 = 3/sin9
AP = 3sin15/sin9 = 4.963465... ( I stored in calculator's memory)
In the right-angled triangle, PBQ
sin24 = PQ/AP
PQ = APsin24 = 2.0188 miles high
Well, first of all, I must say that a weather balloon must feel quite inflated knowing that it caught the attention of not one, but two observers! Let's see if we can figure out its height.
To start, we'll call the height of the balloon "h" (Yes, h stands for height, not helium!). Now, let's do a little triangulation.
From the first observer's perspective, we have an angle of elevation of 15 degrees. Not too shabby! Now, from the second observer's position, the angle of elevation is 24 degrees. The distance between the observers is given as 3 miles, but we won't be pulling any distance-related jokes here.
Let's draw this situation out on a piece of paper (or an imaginary Etch A Sketch, if you prefer). We have two right triangles, each with the same height (h). The first triangle has an angle of 15 degrees, and the second triangle has an angle of 24 degrees. Are you following along? Great!
Now, let's use some trigonometry wizardry. In both triangles, we have the opposite side (which represents the height of the balloon) and the adjacent side (which represents the distance between the observers). We also know the angles. Hocus pocus, let's use tangent!
In the first triangle:
tan(15 degrees) = h / 3 miles
In the second triangle:
tan(24 degrees) = h / 0 miles (because it's a pure vertical line)
Now, let's solve for h. Using some advanced mathematical tools (also known as a calculator), we find that:
h = 3 miles * tan(15 degrees) = 0.79 miles (approximately)
So, the height of the balloon is approximately 0.79 miles. That's quite impressive for a floating rubber duck!
To solve this problem, we will use trigonometry. Let's assign variables to the given information:
Let the height of the balloon be h.
Let the distance between the first observer and the balloon be x.
From the first observer's perspective, we can form a right triangle with the height of the balloon as the opposite side, and the distance x as the adjacent side. We can use the tangent function to express the relationship:
tan(15º) = h / x
From the second observer's perspective, we can form a right triangle with the height of the balloon as the opposite side, and the distance x + 3 miles as the adjacent side. We can use the tangent function again to express the relationship:
tan(24º) = h / (x + 3)
To solve the problem, we need to find the value of h. Let's start by finding the value of x using the first equation:
x = h / tan(15º)
Substitute this value of x in the second equation:
tan(24º) = h / (h / tan(15º) + 3)
Simplify the equation:
tan(24º) = h / (h / tan(15º) + 3)
tan(24º) = h * tan(15º) / (h + 3tan(15º))
Cross-multiply:
h * tan(24º) = h * tan(15º) + 3tan(15º) * h
h * (tan(24º) - tan(15º)) = 3tan(15º) * h
Cancel out the h on both sides:
tan(24º) - tan(15º) = 3tan(15º)
Now, we can solve for h by isolating it on one side:
h = (3tan(15º)) / (tan(24º) - tan(15º))
Using a calculator, we can evaluate the expression:
h ≈ (3 * 0.2679) / (0.4450 - 0.2679)
h ≈ 0.8037 / 0.1771
h ≈ 4.54
Therefore, the balloon is approximately 4.54 miles high.
To find the height of the balloon, we can use trigonometry and set up two right triangles. Let's call the height of the balloon H.
In the first observer's triangle, we have:
Angle of elevation = 15 degrees
Distance from the observer to the balloon = H (height of the balloon)
In the second observer's triangle, we have:
Angle of elevation = 24 degrees
Distance from the second observer to the balloon = H - 3 miles (since the second observer is 3 miles east of the first observer)
Now, let's use the tangent function to solve for the height of the balloon.
In the first observer's triangle:
tan(15 degrees) = H / D1
In the second observer's triangle:
tan(24 degrees) = H / (D1 - 3 miles)
We can rearrange these equations to solve for H:
H = D1 * tan(15 degrees)
H - 3 miles = (D1 - 3 miles) * tan(24 degrees)
Now, let's substitute the values into the equations and solve for H.
H = D1 * tan(15 degrees)
H - 3 miles = (D1 - 3 miles) * tan(24 degrees)
We need the value of D1 to proceed. However, it is not given in the question. Please provide the distance from the first observer to the balloon, and we can calculate the height accordingly.