Problem 28 - Algebra 2

Question

The optimal height \(h\) of the letters of a message printed on pavement is given by the formula \(h=\frac{0.0052 d^{227}}{e} .\) Here \(d\) is the distance of the driver from the letters and \(e\) is the height of the driver's eye above the pavement. All of the distances are in meters. Find \(h\) for the given values of \(d\) and \(e .\) $$d=50 \mathrm{m}, e=2.3 \mathrm{m}$$

Step-by-Step Solution

Answer

The optimal height of the letters for the given values is approximately 0.843 meters.

Step 1: Identify the Given Values

Here the given values are the distance of the driver from the pavement, \(d = 50\) m, and the height of the driver's eye above the pavement, \(e = 2.3\) m.

Step 2: Substitute the Given Values into the Formula

The formula to calculate the optimal height of the letters is \(h=\frac{0.0052 \times d^{227}}{e}\). By substituting \(d = 50\) m and \(e = 2.3\) m into the formula, we get \(h=\frac{0.0052 \times 50^{2.27}}{2.3}\).

Step 3: Calculate the Optimal Height of the Letters

After substituting the given values into the formula, the next step is to perform the calculation in order to determine the value for h. Performing the above calculation gives \(h \approx 0.843\) m.

Key Concepts

Height Calculation

Calculating the optimal height of letters involves understanding a mathematical expression that considers the distance and eye height variables. In the provided exercise, the formula to find the height, \( h \), is given by:
\[ h = \frac{0.0052 \times d^{2.27}}{e} \]
Here, \( d \) is the distance from the driver's eye to the letters, and \( e \) is the height of the driver's eye from the pavement.
When we plug in the specific values—\( d = 50 \) meters and \( e = 2.3 \) meters—we can directly substitute these numbers into the formula. This shows the relationship between these variables and the height of the letters we need to calculate.
Using the formula, all parts need to be calculated step by step, ensuring accuracy. This calculation tells us the best height for the letters so that drivers can read them easily.

Distance and Height Relationships

The interplay between distance and eye height is critical in designing clear and readable messages on the pavement. This relationship is captured in the formula by the variables \( d \) and \( e \).

• **Distance** (\( d \)): The distance from the driver’s eye to the letters is crucial. As the distance increases, the shape and size of the letters need to adjust accordingly, following the power of 2.27 in the formula.
• **Eye Height** (\( e \)): This is the vertical distance from the driver’s eye to the ground. The higher the viewpoint, the lesser the tweak needed in the letter size due to a wider field of view.

These two variables work together to ensure the letters are readable from a given distance. The higher the letters need to be based on the distance and viewing angle, the more the size should compensate for decreasing visual clarity.
This ensures safety and clear communication for drivers who need to read these messages while driving.

Exponential Functions

The exponential function plays a central role in this height calculation, specifically in terms of how size adjustments are made for varying distances. The function \( d^{2.27} \) highlights the non-linear relationship between distance and the optimal height of letters.

• **Exponent Value—2.27**: The exponent value of 2.27 indicates the rate of change for the size of the letters. It shows that as distance grows, the height doesn't just increase linearly but follows a larger scale, hence potentially exponential.
• **Impact of Exponent**: Such an exponent enhances the growth of height significantly compared to if you merely multiplied by distance. It captures the need for larger and larger height adjustments as distances become longer.

Understanding how exponential functions work here is vital. They let designers create signage that maintains readability at significant distances, accounting for how human sight diminishes with distance. Thus, the exponential component ensures that the formula accurately adjusts heights according to these non-linear growth patterns.