Question

A child holds a baloon in her hand at 2 feet. the balloon is attached to a 25 inch string, and the wind blows the baloon so the string makes a angle theta with the verticle. write theta as a function of the height (h) of the balloon from the ground in inches

Answer 1

To find the angle theta as a function of the height h of the balloon from the ground, we use the cosine ratio cosθ = (h - 24) / 25 and solve for θ = cos^-1((h - 24) / 25).

Step-by-step explanation:

To write θ as a function of the height (h) of the balloon from the ground in inches, we use trigonometry and the given length of the string.

Assuming that the string is taut and there are no obstructions, we have a right-angled triangle where the string is the hypotenuse of length 25 inches, the height from the child's hand to the balloon is the opposite side, and the vertical distance from the ground to the child's hand (2 feet, which is 24 inches) is part of the adjacent side.

The total vertical height h from the ground to the balloon thus includes the height of the child's hand and the part of the string that is vertical. Therefore, the remaining length of string making the hypotenuse can be stated as √(252 - (h-24)2).

Then, to find θ, we use the cosine ratio which is adjacent over hypotenuse:

cosθ = (h - 24) / 25

Solving for θ, we get:

θ = cos-1((h - 24) / 25)

This equation allows us to calculate the angle θ for any height h of the balloon from the ground.