Problem:
A hot air balloon rising straight up from a level field is tracked by a range finder located 500 feet from the point of liftoff.
Express the balloon’s height as a function of the angle the line from the range finder to the balloon makes with the ground.
If the sine of the angle is , find the height of the balloon.
a) Given the above problem, provide written responses to address the following points as precisely and thoroughly as possible.
b) Explain the problem in your own words.
c) What mathematical concepts learned in this module apply to this problem?
d) Explain the steps you must take to solve this problem
e) What is the most difficult aspect of solving this problem?
f) Explain exactly what the answer means from a mathematical perspective.
This post was also placed at freemathhelp, on the calculus board with titled, "Calculus I".
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a) Here are the precise and thorough responses:
b) The problem is asking us to express the height of a hot air balloon as a function of the angle that the line from the range finder to the balloon makes with the ground. We are also given the sine of the angle and asked to find the height of the balloon.
c) The mathematical concepts that apply to this problem are trigonometry, specifically the concept of sine. Sine is a trigonometric function that relates the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse.
d) To solve this problem, we can use the definition of sine. The sine of an angle is equal to the length of the side opposite the angle divided by the length of the hypotenuse. In this case, the side opposite the angle is the height of the balloon and the hypotenuse is the distance between the range finder and the point of liftoff.
Let's denote the height of the balloon as h and the angle as θ. According to the problem, the sine of θ is given, so we can write the equation:
sin(θ) = h / 500
To solve for h, we can rearrange the equation:
h = 500 * sin(θ)
Now we can substitute the given value of sine and calculate the height of the balloon.
e) The most difficult aspect of solving this problem may be understanding and applying the concept of sine and how it relates to the given information. It requires a clear understanding of the definition of sine and how to rearrange the equation to solve for the unknown variable.
f) The answer to this problem, h = 500 * sin(θ), represents the height of the hot air balloon. The height is a function of the angle that the line from the range finder to the balloon makes with the ground. As the angle increases, the height of the balloon also increases. The sine function allows us to relate the angle to the height.