(Modeling) Distance from the Origin of the Nile River The Nile River in Africa is about 4000 mi long. The Nile begins as an outlet of Lake Victoria at an altitude of 7000 ft above sea level and empties into the Mediterranean Sea at sea level (0 ft). The distance from its origin in thousands of miles is related to its height above sea level in thousands of feet (x) by the following formula. 7 - x Distance = ——————— 0.639x + 1.75 For example, when the river is at an altitude of 600 ft, x = 0.6 (thousand), and the distance from the origin is 7 - 0.6 Distance = ————————— ≈ 3, which represents 3000 mi. 0.639(0.6) + 1.75 (Data from The World Almanac and Book of Facts.) What is the distance from the origin of the Nile when the river has an altitude of 7000 ft?

Verified step by step guidance

1. Identify the altitude of the river in thousands of feet: 7000 ft corresponds to x = 7.
2. Substitute x = 7 into the formula for distance: $\text{Distance}=\frac{7-x}{0.639x+1.75}$.
3. Simplify the numerator: $7-7=0$.
4. Substitute the simplified numerator into the formula: $\text{Distance}=\frac{0}{0.639(7)+1.75}$.
5. Recognize that any number divided by zero is zero, so the distance from the origin is 0 miles.

Key Concepts

Function and Variables

In mathematics, a function is a relation that assigns each input exactly one output. In this context, the distance from the origin of the Nile River is expressed as a function of its altitude above sea level. Understanding how to interpret the variables in the formula is crucial for calculating the distance based on different altitudes.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve for unknowns. In this problem, substituting the altitude into the given formula requires careful handling of fractions and operations to arrive at the correct distance. Mastery of these skills is essential for accurately solving equations in trigonometry and other mathematical fields.

Units of Measurement

Understanding units of measurement is vital in this problem, as the distance is given in miles and the altitude in feet. The formula uses thousands of miles and feet, which necessitates converting values appropriately to maintain consistency. Recognizing how to convert between different units ensures accurate calculations and interpretations of the results.