Problem 96 - Algebra and Trigonometry with Analytic Geometry

Question

Viewing distance On a clear day, the distance \(d\) (in miles) that can be seen from the top of a tall building of height \(h\) (in feet) can be approximated by \(d=1.2 \sqrt{h}\). Approximate the distance that can be seen from the top of the Chicago Sears Tower, which is 1454 feet tall.

Step-by-Step Solution

Answer

The distance visible from the Sears Tower is approximately 45.7 miles.

Step 1: Identify the given information

We are given the height of the Sears Tower, which is \( h = 1454 \) feet. We need to calculate the distance \( d \) that can be seen from the top of the tower using the provided formula.

Step 2: Write the formula

The distance \( d \) that can be seen from the height \( h \) is given by the formula \( d = 1.2 \sqrt{h} \).

Step 3: Substitute the given height into the formula

Substitute \( h = 1454 \) into the formula: \[ d = 1.2 \sqrt{1454} \].

Step 4: Calculate the square root

First, find the square root of 1454: \( \sqrt{1454} \approx 38.11 \).

Step 5: Multiply to find the distance

Multiply the result by 1.2: \( d = 1.2 \times 38.11 \approx 45.732 \).

Step 6: Conclude the calculation

Thus, the approximate distance that can be seen from the top of the Sears Tower is about 45.7 miles.

Key Concepts

Square Root

The square root is a fundamental mathematical concept. It is the number which, when multiplied by itself, gives the original number. In our exercise, we are given the height of the Sears Tower, at 1454 feet, and we need to calculate the square root of this to find the distance visible from the tower. An easy way to think of a square root is that it's the opposite action of squaring a number. For example, since 5 times 5 equals 25, the square root of 25 is 5.
In calculations, finding the square root manually can be tricky, but it's important for solving equations like the one in our exercise:

• Start by identifying whether you are dealing with a perfect square or not. In this case, 1454 is not a perfect square.
• Use a calculator for non-perfect squares to get precise results. For 1454, we approximate it at around 38.11.

This number is then used in further calculations, like in our provided exercise formula.

Heights and Distances

In mathematics and physics, heights and distances commonly appear when calculating how far one can see from atop a tall structure. Understanding the relationship between these two can provide a deeper insight into practical applications such as surveying and architecture.
The formula provided, \[ d = 1.2 \sqrt{h} \]assumes a direct correlation between the height and the distance in clear conditions. Building height and distance do not increase at the same rate, due to how the eyeball perceives distance in relation to height. Thus, such formulas help us make precise estimates.
An important thing to remember is that this calculation

• Does not account for obstructions like other buildings or natural formations.
• Is based on an approximation for Earth’s curvature when not indicated otherwise.

This relationship demonstrates how height can give us a powerful vantage point, effectively increasing the visible range.

Approximation

Approximation is a valuable technique in mathematics, used to find a value that is close enough to the right answer for practical purposes. For instance, in our exercise calculating how far one can see from the Sears Tower involves approximating the square root of 1454.
Approximating numbers can often make complex calculations manageable. Here’s how it plays a role in our example:

• Simplifying Difficult Problems: Instead of finding the exact decimal value of \( \sqrt{1454}\), we approximate it to 38.11.
• Converting into a Usable Form: This makes the multiplication process straightforward and eliminates excessive precision that doesn't significantly alter results.
• Estimating Real-Life Scenarios: Such techniques are vital in scenarios where exact values are cumbersome or unnecessary, like estimating visible distance from a height.

Using approximation allows problem-solving to remain effective, practical, and relevant to real-life scenarios.