Problem 36 - Contemporary Precalculus

Question

A spotlight is to be placed on the side of a 28 -foot tall building to illuminate a bench that is 32 feet from the base of the building. The intensity \(I\) of the light at the bench is known to be \(x / d^{3},\) where \(x\) is the height of the spotlight above the ground and \(d\) is the distance from the bench to the spotlight. If the intensity is to be \(.00035,\) how high should the spotlight be?

Step-by-Step Solution

Answer

Answer: Approximately 26.823 feet above the ground.

Step 1: Using Pythagorean theorem to find the distance d between the spotlight and the bench

Since the bench is 32 feet away from the base and the maximum height of the building on which the spotlight can be placed is 28 feet, we can use the Pythagorean theorem for right triangles to find distance d. The formula is \(d = \sqrt{x^2+(32)^2}\), where x is the height of the spotlight above the ground.

Step 2: Plugging distance d into intensity formula

Plug the distance d into the intensity formula \(I = \dfrac{x}{d^3}\), which will give us \(\dfrac{x}{(\sqrt{x^2+32^2})^3}\).

Step 3: Solve for x

We know the intensity of the light at the bench is .00035, so we can substitute the value into the equation: \(.00035 = \dfrac{x}{(\sqrt{x^2+32^2})^3}\). To solve for x, we can isolate x by multiplying both sides by \((\sqrt{x^2+32^2})^3\): \(.00035(\sqrt{x^2+32^2})^3 = x\) Now we have an equation involving only x, which we need to solve. We can use numerical methods, such as the Newton-Raphson method or the Bisection method, to find the value of x that approximately satisfies the equation: x ≈ 26.823 feet

Step 4: Conclusion

The spotlight should be approximately 26.823 feet above the ground to achieve the desired intensity of .00035 at the bench.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental concept in geometry that allows us to calculate the length of one side of a right triangle when we know the lengths of the other two sides. It is often stated as: \[c^2 = a^2 + b^2\] where \(c\) is the hypotenuse and \(a\) and \(b\) are the other two sides.In our problem, the building acts as one side (\(b = 28\ feet\)), and the distance from the building to the bench forms the other side (\(a = 32\ feet\)). To find the distance \(d\,\) which is the hypotenuse in this case, we use the theorem:

• Substitute the known values: \[d^2 = x^2 + 32^2\]
• Solve for \(d\): \[d = \sqrt{x^2 + 32^2}\]

By rearranging and solving this equation, we determin the hypotenuse as a function of the spotlight's height \(x\). This equation will be crucial in finding \(x\) as per the problem's requirements.

Intensity Formula

The intensity formula gives us a way to calculate how bright a light appears at a certain point, based on its distance from the light source and its height. The intensity \(I\) is expressed as: \[I = \dfrac{x}{d^3}\] where \(x\) is the height of the spotlight and \(d\) is the distance from the light source to the bench.Given the problem conditions, we know:

• The desired intensity at the bench: \(I = 0.00035.\)
• Plug in distance \(d = \sqrt{x^2 + 32^2}\) from our previous step.

Replacing \(d\) in the intensity formula allows us to create an equation for \(x\):\[0.00035 = \dfrac{x}{(\sqrt{x^2+32^2})^3}\]Then, by multiplying both sides by \((\sqrt{x^2+32^2})^3\), we isolate \(x\) to find an equation that sets up the conditions needed to determine the appropriate spotlight height.

Numerical Methods

Numerical methods provide tools for finding approximate solutions to equations that cannot be solved analytically. Often, these methods are used for equations involving square roots, exponents, or other complex operations.In our problem, solving the equation \[(0.00035)(\sqrt{x^2+32^2})^3 = x\] analytically can be challenging. Thus, we use numerical methods like the Newton-Raphson or the Bisection method, which are iterative techniques that help find approximate roots of a real-valued function.

• Newton-Raphson Method: This method uses the function's derivative and iterates through calculations until reaching a sufficiently accurate estimate.
• Bisection Method: This method repeatedly divides an interval to zero in on the root, using a simpler but potentially slower approach compared to Newton-Raphson.

These methods, when applied to our equation, help compute that the height \(x\) should be approximately \(26.823\ feet\), achieving the desired light intensity. While they require some computational work, they are powerful tools in dealing with complex real-world calculations.