On A Morning Of A Day When The Sun Will Pass Directly Overhead, The Shadow Of A 72-ft Building On Level Ground Is 21 Ft Long, As Shown In The Figure. At The Moment In Question, The Angle The Sun Makes With The Ground Is Increasing At The Rate Of

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Introduction

In this article, we will delve into a problem involving trigonometry and the concept of shadow length. We will explore how to calculate the rate at which the angle of the sun is increasing, given the length of the shadow of a building and the height of the building itself.

The Problem

On a morning of a day when the sun will pass directly overhead, the shadow of a 72-ft building on level ground is 21 ft long, as shown in the figure. At the moment in question, the angle the sun makes with the ground is increasing at a certain rate. Our goal is to determine this rate.

Visualizing the Problem

To begin, let's visualize the problem. We have a building of height 72 ft, and its shadow is 21 ft long. The angle the sun makes with the ground is increasing at a certain rate. We can represent this situation using a right-angled triangle, with the building as the opposite side, the shadow as the adjacent side, and the line of sight to the sun as the hypotenuse.

Using Trigonometry to Solve the Problem

We can use trigonometry to solve this problem. Specifically, we can use the tangent function, which relates the opposite side (the building) to the adjacent side (the shadow). The tangent function is defined as:

tan(θ) = opposite side / adjacent side

In this case, the opposite side is the height of the building (72 ft), and the adjacent side is the length of the shadow (21 ft). We can plug these values into the tangent function to get:

tan(θ) = 72 ft / 21 ft

Simplifying the Equation

To simplify the equation, we can divide both sides by 21 ft:

tan(θ) = 72 ft / 21 ft tan(θ) = 3.43

Finding the Angle

Now that we have the value of the tangent function, we can use it to find the angle θ. We can use the inverse tangent function (also known as the arctangent function) to do this:

θ = arctan(3.43)

Calculating the Rate of Change

Now that we have the angle θ, we can calculate the rate at which it is increasing. We are given that the angle is increasing at a certain rate, which we can represent as a derivative. The derivative of the angle with respect to time is:

dθ/dt = ?

To find this derivative, we can use the chain rule and the fact that the tangent function is the derivative of the arctangent function:

dθ/dt = d/dt (arctan(3.43)) dθ/dt = (1 / (1 + (3.43)^2)) * d/dt (3.43)

Simplifying the Derivative

To simplify the derivative, we can plug in the value of the derivative of the arctangent function:

dθ/dt = (1 / (1 + (3.43)^2)) * 0 dθ/dt = 0

Conclusion

In this article, we have used trigonometry to solve a problem involving the length of a shadow and the rate at which the angle of the sun is increasing. We have found that the angle is increasing at a rate of 0 radians per second.

Key Takeaways

  • The tangent function relates the opposite side to the adjacent side in a right-angled triangle.
  • The inverse tangent function (arctangent function) is used to find the angle given the value of the tangent function.
  • The derivative of the angle with respect to time is used to find the rate at which the angle is increasing.

Further Reading

For further reading on trigonometry and the tangent function, we recommend the following resources:

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline

References

  • Corral, M. (2018). Trigonometry. CreateSpace Independent Publishing Platform.
  • Spivak, M. (2008). Calculus. Publish or Perish, Inc.
  • Kline, M. (1972). Mathematics for the Nonmathematician. Dover Publications.
    Frequently Asked Questions: Understanding the Problem of Shadow Length ====================================================================

Q: What is the problem about?

A: The problem is about finding the rate at which the angle of the sun is increasing, given the length of the shadow of a building and the height of the building itself.

Q: What is the given information?

A: The given information is that the shadow of a 72-ft building on level ground is 21 ft long, and the angle the sun makes with the ground is increasing at a certain rate.

Q: What is the goal of the problem?

A: The goal of the problem is to determine the rate at which the angle of the sun is increasing.

Q: What is the relevance of the tangent function in this problem?

A: The tangent function is used to relate the opposite side (the building) to the adjacent side (the shadow) in a right-angled triangle.

Q: How is the angle θ related to the tangent function?

A: The angle θ is related to the tangent function by the equation:

tan(θ) = opposite side / adjacent side

Q: What is the value of the tangent function in this problem?

A: The value of the tangent function is 3.43.

Q: How is the angle θ calculated?

A: The angle θ is calculated using the inverse tangent function (arctangent function):

θ = arctan(3.43)

Q: What is the derivative of the angle θ with respect to time?

A: The derivative of the angle θ with respect to time is:

dθ/dt = d/dt (arctan(3.43)) dθ/dt = (1 / (1 + (3.43)^2)) * d/dt (3.43)

Q: What is the final answer to the problem?

A: The final answer to the problem is that the angle θ is increasing at a rate of 0 radians per second.

Q: What are some key takeaways from this problem?

A: Some key takeaways from this problem are:

  • The tangent function relates the opposite side to the adjacent side in a right-angled triangle.
  • The inverse tangent function (arctangent function) is used to find the angle given the value of the tangent function.
  • The derivative of the angle with respect to time is used to find the rate at which the angle is increasing.

Q: What are some further reading resources for this topic?

A: Some further reading resources for this topic are:

  • "Trigonometry" by Michael Corral
  • "Calculus" by Michael Spivak
  • "Mathematics for the Nonmathematician" by Morris Kline

Q: What are some real-world applications of this problem?

A: Some real-world applications of this problem include:

  • Calculating the angle of the sun for solar panel installations
  • Determining the height of a building given the length of its shadow
  • Understanding the behavior of light and shadows in various environments

Q: What are some common misconceptions about this problem?

A: Some common misconceptions about this problem include:

  • Assuming that the angle of the sun is always increasing at a constant rate
  • Failing to account for the effects of atmospheric refraction on the angle of the sun
  • Not considering the importance of the tangent function in relating the opposite side to the adjacent side in a right-angled triangle.