The angles of elevation of a balloon directly above a straight road, from two points of the ground on opposite sides of the balloon, are 78¡ã15' and 59¡ã47'. If the points are 5000ft apart, what is the height of the balloon?

Try to answer

Well, if we're dealing with balloons, I must say this question really takes the air out of me! But fear not, I'm here to float you the answer.

To find the height of the balloon, we need to use some trigonometry. Now, we have two angles of elevation: 78°15' and 59°47'. Let's use the smaller angle for our calculations.

First, let's convert the angles to decimal form. 78°15' becomes 78.25° and 59°47' becomes 59.783°.

Now, let's draw a diagram. We have a straight road, and the balloon is directly above it. The two points of observation create a triangle with the balloon at the top. The 5000ft distance between the two points is the base of the triangle.

Using tangent (tan), we can calculate the height (h) of the balloon using the formula:

h = distance * tan(angle)

For our calculations:

h = 5000ft * tan(59.783°)

Calculating this, we find that the height of the balloon is approximately 9381.23 feet.

So there you have it! The height of the balloon is around 9381.23 feet. Just remember, my answers may be funny, but I always aim to bring a smile to your face!

To find the height of the balloon, we can use the property of tangent in a right-angled triangle.

Let's denote the height of the balloon as 'h' and the distance from one of the points to the balloon as 'x'.

From the given information, we have two right-angled triangles for each point of observation:

Triangle 1:
Angle of elevation = 78°15'
Length of adjacent side = x
Length of opposite side = h

Triangle 2:
Angle of elevation = 59°47'
Length of adjacent side = 5000 - x
Length of opposite side = h

Using the tangent function, we can set up the following equations:

tan(78°15') = h/x
tan(59°47') = h/(5000 - x)

Now, let's solve for 'h':

tan(78°15') = h/x
tan(78°15') * x = h [Equation 1]

tan(59°47') = h/(5000 - x)
tan(59°47') * (5000 - x) = h [Equation 2]

Setting the values into the calculator, we get:
tan(78°15') ≈ 4.00479
tan(59°47') ≈ 1.71024

Plugging these values into Equations 1 and 2:

4.00479 * x = h [Equation 1]
1.71024 * (5000 - x) = h [Equation 2]

Now, we have a system of two equations with two unknowns (h and x). Let's solve this system:

From Equation 1, we get h = 4.00479 * x
Plugging this value into Equation 2:

1.71024 * (5000 - x) = 4.00479 * x

Simplifying the equation:

8551.2 - 1.71024x = 4.00479x
8551.2 = 5.71403x
x ≈ 1498.88

Now, we have the value of 'x'. Let's find the value of 'h' by substituting 'x' into Equation 1:

h = 4.00479 * x
h = 4.00479 * 1498.88
h ≈ 5,997.67

The height of the balloon is approximately 5,997.67 feet.

To find the height of the balloon, we can use the concept of trigonometry. Let's assume that the height of the balloon is h.

First, let's draw a diagram to visualize the problem. We have a straight road, and on opposite sides of the balloon, there are two points from which the angles of elevation are given. Let's call these points A and B. The balloon is directly above the road at point C.

C
/|
/ |
h / |
/ |
/ |
/ |
/a A\
/_ _ _ _\

Point A is higher than the road, and the angle of elevation from point A to the balloon is 78°15'.

Point B is on the same line as the road, and the angle of elevation from point B to the balloon is 59°47'.

Let's calculate the height of the balloon using trigonometry.

First, we need to find the distance from Point A to Point C along the road. Let's call this distance d.

Using the trigonometric relation:
tan(angle of elevation) = (height of balloon) / (distance from Point A to C)

For Point A, we have:
tan(78°15') = h / d

We can write this as:
h = d * tan(78°15')

Now, let's find the distance from Point B to Point C along the road. Since B is on the same line as the road, this distance is equal to 5000ft.

Next, let's use the relation for Point B:
tan(59°47') = h / 5000

We can write this as:
h = 5000 * tan(59°47')

Now we have two equations for the height of the balloon in terms of d and h. We can equate these two equations to eliminate h and solve for d.

d * tan(78°15') = 5000 * tan(59°47')

Now, we can rearrange the equation to solve for d:
d = (5000 * tan(59°47')) / tan(78°15')

By plugging in the values and evaluating this expression, we can find the distance d.

Finally, we have the distance d. To find the height of the balloon, we can substitute the value of d into any of the equations we derived earlier.
Let's use the equation:
h = d * tan(78°15')

By substituting the value of d, we can calculate the height of the balloon.

draw a diagram.

Review your basic trig functions.
Now it is clear that if the height is h ft,

h cot 78°15' + h cot 59°47' = 500