Two stations, A and B, are located six miles apart on a railroad. The rates of cartage of coal are 50 cents per ton per mile from A and 75 cents per ton per mile from B. At a certain consumer's home, located on the railroad between A and B, the cost for cartage is the same whether the coal is delivered from A or from B. Find the distance from this home to A.
a. 3 miles
b. 3 1/2 miles
c. 3 3/5 miles
d. 4 miles
e. 4 1/2 miles
please answer and explain
cost is rate * distance. If the distance from home to A is x, then from home to B is 6-x. So, since the costs are the same,
50x = 75(6-x)
50x = 450-75x
125x = 450
x = 3.6
So, (C)
To solve this problem, we can set up an equation based on the given information. Let's assume that the distance from the consumer's home to A is x miles. Therefore, the distance from the consumer's home to B would be (6 - x) miles.
The cost of cartage of coal from A is $0.50 per ton per mile, and the distance from A to the consumer's home is x miles. So, the cost of cartage from A would be $0.50 * x.
Similarly, the cost of cartage of coal from B is $0.75 per ton per mile, and the distance from B to the consumer's home is (6 - x) miles. Hence, the cost of cartage from B would be $0.75 * (6 - x).
Since the question states that the cost for cartage is the same whether the coal is delivered from A or B, we can set up an equation:
$0.50 * x = $0.75 * (6 - x)
Let's solve the equation:
0.50x = 0.75(6 - x)
0.50x = 4.5 - 0.75x
0.50x + 0.75x = 4.5
1.25x = 4.5
x = 4.5 / 1.25
x = 3.6
Therefore, the distance from the consumer's home to A is approximately 3.6 miles.
Among the given options, the closest one is 3 3/5 miles (c), which is the answer to the problem.