A surveyor discovers that the total combined length of all roads, in miles (mi), of a particular city was 500 at the beginning of 1950 and increased by 125mi every ten years. Additionally, the number of cars driven in the city was 50,000 at the beginning of 1950 and increased by 20 percent every ten years. What was the difference between the number cars driven in the city per mile of city road at the beginning of 1950 and at the beginning of 1990 ? Choose 1 answer: (A) 3.68 (B) 10 (C) 107.36 (D) 103.68

Question

A surveyor discovers that the total combined length of all roads, in miles (mi), of a particular city was 500 at the beginning of 1950 and increased by  125mi every ten years. Additionally, the number of cars driven in the city was 50,000 at the beginning of 1950 and increased by 20 percent every ten years. What was the difference between the number cars driven in the city per mile of city road at the beginning of 1950 and at the beginning of 1990 ? Choose 1 answer: (A) 3.68 (B) 10 (C) 107.36 (D) 103.68

Bytelearn · Accepted Answer

Calculate road length in 1990: Let's calculate the length of all roads in the city at the beginning of 1990. The length of roads in 1950: 500 mi Increase every ten years: 125 mi Number of ten-year periods from 1950 to 1990: 4 (1950-1960, 1960-1970, 1970-1980, 1980-1990) Total increase in road length by 1990: 125 mi * 4 = 500 mi Total length of roads in 1990: 500 mi (initial) + 500 mi (increase) = 1000 miCalculate number of cars in 1990: Now let's calculate the number of cars in the city at the beginning of 1990. The number of cars in 1950: 50,000 Percentage increase every ten years: 20% To calculate the increase, we use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the amount after time t, P is the principal amount (initial amount), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years. Since the increase is every ten years, we treat it as compounded once every ten years (n=1), and the rate is 20% or 0.20. Number of ten-year periods from 1950 to 1990: 4 Number of cars in 1990: 50,000 * (1 + 0.20)^4Formula for compound interest: Calculating the number of cars in 1990 using the formula: Number of cars in 1990 = 50,000 * (1 + 0.20)^4 Number of cars in 1990 = 50,000 * (1.20)^4 Number of cars in 1990 = 50,000 * 2.0736 Number of cars in 1990 = 103,680Calculate number of cars in 1990 using the formula: Now we will calculate the number of cars per mile of road at the beginning of 1950 and at the beginning of 1990. Number of cars per mile in 1950: 50,000 cars / 500 mi = 100 cars/mi Number of cars per mile in 1990: 103,680 cars / 1000 mi = 103.68 cars/miCalculate number of cars per mile in 1950 and 1990: Finally, we find the difference between the number of cars per mile of road in 1950 and 1990. Difference = Number of cars per mile in 1990 - Number of cars per mile in 1950 Difference = 103.68 cars/mi - 100 cars/mi Difference = 3.68 cars/mi

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