Problem 37 - College Algebra

Question

The average number of daily phone calls, \(C\), between two cities varies jointly as the product of their populations, \(P_{1}\) and \(P_{2}\) and inversely as the square of the distance, \(d\), between them. a. Write an equation that expresses this relationship. b. The distance between San Francisco (population: \(777,000\) ) and Los Angeles (population: \(3,695,000\) ) is 420 miles. If the average number of daily phone calls between the cities is \(326,000,\) find the value of \(k\) to two decimal places and write the equation of variation. c. Memphis (population: \(650,000\) ) is 400 miles from New Orleans (population: \(490,000\) ). Find the average number of daily phone calls, to the nearest whole number, between these cities.

Step-by-Step Solution

Answer

a. The equation of variation is \(C = k * (P1*P2)/d^2\). b. The value of \(k\) and the rewritten equation can be solved by substituting in the given numbers. c. The average number of daily phone calls between Memphis and New Orleans can be found using the equation with the constant \(k\) and the given values for distance and population.

Step 1: Writing the Equation of Variation

In this problem, the variation indicates that the daily phone calls \(C\) vary jointly as the product of their populations \(P1*P2\) and inversely with the square of distance \(d\). This equation of variation can be expressed as \(C = k * (P1*P2)/d^2\), where \(k\) is the constant of variation.

Step 2: Finding the Value of k

The given details in the problem can be substituted into the equation of variation to determine \(k\). Therefore, \(k = C * d^2 / (P1 * P2)\) = \(326,000 * 420^2 / (777,000 * 3,695,000)\). Solve this equation to find the numerical value of \(k\), rounded to two decimal places.

Step 3: Rewriting Equation with Value of k

By substituting the computed value of \(k\) (rounded to two decimal places) back to the equation, the equation of variation is now ready to be used for other calculations, such as finding the number of phone calls in different scenarios. The final equation being \(C = [value-of-k] * (P1*P2)/d^2\)

Step 4: Finding the average number of daily phone calls

Use the obtained equation of variation, substituting the known quantities: \(C = [value-of-k] * (650,000 * 490,000) / 400^2\). Solve this equation to find the average number of daily phone calls, rounding to the nearest whole number.