Problem 92 - Precalculus Mathematics for Calculus

Question

Suppose that the graph of \(y=2^{x}\) is drawn on a coordinate plane where the unit of measurement is an inch. (a) Show that at a distance 2 ft to the right of the origin the height of the graph is about 265 mi. (b) If the graph of \(y=\log _{2} x\) is drawn on the same set of axes, how far to the right of the origin do we have to go before the height of the curve reaches 2 ft?

Step-by-Step Solution

Answer

(a) 265 miles; (b) 1,398,101 feet to the right for 2 feet height.

Step 1: Convert feet to inches for part (a)

To solve part (a), we first need to convert feet to inches. Since there are 12 inches in a foot, 2 feet is equal to \[2 \times 12 = 24\] inches.

Step 2: Find the height of the graph for part (a)

For the graph of \(y = 2^x\), the height at \(x = 24\) inches is \[y = 2^{24}\].Computing this, we get:\[2^{24} = 16,777,216\].Since the unit of measurement is in inches, the height in inches is 16,777,216.

Step 3: Convert inches to miles for part (a)

We need to convert 16,777,216 inches to miles, knowing that 1 mile is 63,360 inches. Doing the division gives us:\[\frac{16,777,216}{63,360} \approx 264.7\] miles.So, the height of the graph is about 265 miles.

Step 4: Convert feet to inches for part (b)

In part (b), we're looking for the \(x\) value on the graph \(y = \log_2 x\), where the output is 2 feet, which we convert to inches:\[2 \times 12 = 24\] inches.

Step 5: Solve for x in part (b)

Since we need \(y = \log_2 x\) to be 24 inches, we set up the equation:\[\log_2 x = 24\].To solve for \(x\), we can exponentiate both sides using base 2:\[x = 2^{24}\] which is \(x = 16,777,216\).

Step 6: Calculate the distance to the right for part (b)

The \(x\) value we found, 16,777,216, is in inches because it's \(x\) on the graph where the height in inches is 24. This means we need to go 16,777,216 inches to the right of the origin.

Step 7: Convert inches to feet for part (b)

To convert inches to feet, divide by 12:\[\frac{16,777,216}{12} \approx 1,398,101\] feet.Thus, the curve reaches a height of 2 feet at approximately 1,398,101 feet to the right of the origin.

Key Concepts

Graphing Exponential Functions

Exponential functions are intriguing because they grow really fast. When you graph the exponential function, say, \( y = 2^x \), for different values of \( x \), you notice the graph rises quickly from left to right. This rapid increase is a hallmark of exponential functions.
To sketch the graph, you typically start by plotting a few key points:

• At \( x = 0 \), \( y = 2^0 = 1 \).
• At \( x = 1 \), \( y = 2^1 = 2 \).
• At \( x = 2 \), \( y = 2^2 = 4 \).
• The graph continues to grow exponentially for greater \( x \) values.

As shown in the exercise, when you move 24 inches (i.e., 2 feet) to the right, the graph's height at this point, \( x = 24 \), becomes whopping 16,777,216 inches. If we convert this height to miles, it measures an incredible 265 miles! This example highlights the extraordinary growth rate of exponential functions.

Logarithmic Scale

Logarithmic functions, like \( y = \log_2 x \), are the opposite of exponential functions. They grow slowly and can be used to measure scale in a very different way than their exponential counterparts. The logarithmic scale is helpful when you deal with numbers that span a large range, compressing those differences.

• At \( y = 1 \), the value of \( x \) for \( y = \log_2 x \) is 2.
• At \( y = 2 \), \( x = 4 \).
• As \( y \) increases further, \( x \) grows quickly, demonstrating a logarithmic growth.

In our exercise, we need the graph to reach a height of 2 feet, equivalent to 24 inches. Solving the equation \( \log_2 x = 24 \) gives us \( x = 16,777,216 \). This illustrates that even with a relatively small height (2 feet), the logarithmic scale requires a vast \( x \)-distance to reach it, specifically about 1,398,101 feet.

Conversion of Units

Understanding unit conversion is crucial in many mathematical problems, including this exercise. Units allow us to measure quantities and convert them to different scales. Let's break down the conversions used here:

• Feet to Inches: Since 1 foot contains 12 inches, to convert feet to inches, multiply by 12. Thus, 2 feet becomes \( 2 \times 12 = 24 \) inches.
• Inches to Miles: Knowing that 1 mile equals 63,360 inches helps us convert larger measures. For instance, converting 16,777,216 inches into miles involves dividing by 63,360, yielding approximately 265 miles.
• Inches to Feet: For larger lengths, convert inches back to feet by dividing by 12. So, 16,777,216 inches becomes approximately 1,398,101 feet.

These conversions not only ease our calculations but also help visualize the scale of our results in practical terms.