Problem 145 - Precalculus Essentials

Question

A city's tall buildings and narrow streets reduce the amount of sunlight. If \(h\) is the average height of the buildings and \(w\) is the width of the street, the angle of elevation from the street to the top of the buildings is given by the trigonometric equation $$\tan \theta=\frac{h}{w}$$ A value of \(\theta=63^{\circ}\) can result in an \(85 \%\) loss of illumination. Some people experience depression with loss of sunlight. Determine whether such a person should live on a city street that is 80 feet wide with buildings whose heights average 400 feet. Explain your answer and include \(\theta,\) to the nearest degree, in your argument.

Step-by-Step Solution

Answer

After performing the calculations, the obtained \(\theta\) value should be evaluated and if it is greater than 63^{\circ}, the street might not be suitable for the person due to possible depression resulting from significant loss of illumination. Otherwise, if \(\theta\) is less or equal to 63^{\circ}, the person should consider it habitable.

Step 1: Understanding the Problem

The problem involves the usage of trigonometry to find the angle of elevation between the street and the top of the buildings so we can assess the amount of sunlight reaching the street. The function used is given as \(\tan\theta = \frac{h}{w}\), where \(h\) represents the height of the buildings, \(w\) is the width of the street and \(\theta\) is the angle of elevation.

Step 2: Identify given Values

Here, it is given that the width of the street \(w = 80\) feet and height of the buildings \(h = 400\) feet. Using these values we can find the angle of elevation \(\theta\).

Step 3: Compute the Tangent of the Angle

Substitute the given values into the equation, it becomes \(\tan\theta = \frac{400}{80}\) which involves the computation of \(\tan^{-1}\) to get the angle \(\theta\). That is, \(\theta = \tan^{-1} (\frac{400}{80})\).

Step 4: Compare the Angle

Finally, compare the found value of \(\theta\) with 63^{\circ}. If the computed value is greater than 63^{\circ}, it means the street has more than 85% loss of illumination which might not be suitable for the person since it could result in depression. Otherwise, if the value is less than or equal to 63^{\circ}, the street should be considered habitable for that person.

Key Concepts

Angle of Elevation

When you stand on the street and look up at the top of a building, you create an angle with your line of sight. This angle is called the angle of elevation. It is the angle between the horizontal line to the building top from your eyes. In trigonometry, understanding angles of elevation helps solve real-world problems like figuring out how much sunlight shines down narrow city streets. In this exercise, we have buildings 400 feet tall and streets 80 feet wide. The angle, \(\theta\), tells us how steeply you look upwards to see the building top. Calculating the angle of elevation allows us to determine how sunlight behaves in narrow urban environments and decide if locations are suitable for people who need light exposure.

Tangent Function

The tangent function, used widely in trigonometry, relates to angles of right triangles. In this problem, the tangent of the angle of elevation \(\theta\) is calculated using the formula \(\tan\theta=\frac{h}{w}\). This equation engages the height of the building ( \(\h\) ) and the street's width ( \(\w\) ) to derive \(\theta\).

• If you know the building height and street width, you use those values in the formula to find \(\theta\).
• For our case, substituting \(\h=400\) feet and \(\w=80\) results in \(\tan\theta=\frac{400}{80}\), hence the tangent of our desired angle.

After calculating the tangent, the inverse tangent function, \(\tan^{-1}\) , or arctan, retrieves the angle \(\theta\) using common calculators or mathematical functions. Knowing \(\theta\) helps us understand if sunlight is enough or reduced to a level causing concern.

Illumination Loss

Illumination loss in city streets largely depends on the angle \(\theta\), which helps calculate the amount of sunlight reaching down alleys and roads. When \(\theta=63^{\circ}\) means sunlight diminishes significantly. Here, reduction reaches 85%, indicating buildings may severely block light. Such high-level sunlight loss affects environments and people living in them. Particularly for individuals sensitive to sunlight or prone to seasonal affective disorders, living in such shaded streets may not be advisable.Steps to determine illumination loss include:

• Computing \(\theta\)
• Comparing with 63 degrees
• Deciding if the environment satisfactorily receives sunlight

In the exercise, with a calculated \(\theta\) > 63 degrees, over 85% illumination loss is likely, raising red flags if sunlight is desired for well-being.