a hot air balloon climbs continuously along a 30° angle to a height of 5,000 feet. To the nearest tenth of a foot, how far has the balloon traveled to reach 5,000 feet? Draw a sketch and then solve.

I assume you want the distance traveled along the hypotenuse, d, then

sin30° = 5000/d
d = 5000/sin30
d = ...

if you want the horizontal distance traveled , x

tan30 = 5000/x
x = 5000/tan30
= ...

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To solve this problem, we can use trigonometry, specifically the sine function, to calculate the horizontal distance traveled by the balloon.

First, let's draw a sketch to better understand the situation:

/|
/ |
/ |
/ |
/ |
5,000ft / | Vertical Distance
/ |
/ |
/ |
/ |
/ |
/ θ | Horizontal Distance
/_____________

In the diagram, the vertical line represents the balloon's climb of 5,000 feet, and the angle θ is the angle of ascent, which is given as 30°.

Now, let's apply the concept of trigonometry:

In a right-angled triangle, the sine of an angle is the ratio of the opposite side (in this case, the vertical distance) to the hypotenuse (the distance traveled by the balloon).

Using the formula sin(θ) = vertical distance / hypotenuse, we can rearrange the formula to solve for the hypotenuse (horizontal distance):

hypotenuse = vertical distance / sin(θ)

Given that the vertical distance is 5,000 feet and the angle θ is 30°, we can substitute these values into the formula:

hypotenuse = 5,000 ft / sin(30°)

Now, let's evaluate the expression:

hypotenuse = 5,000 ft / 0.5 (sin(30°) is 0.5)

hypotenuse = 10,000 ft

Therefore, the balloon has traveled approximately 10,000 feet horizontally to reach a height of 5,000 feet.