A man drove his automobile to New York, a distance of 550 miles, at a certain average rate. He returned from New York at the rate of 11 mph more. The entire trip required 22.5 hours. What were his rates returning and going?
first rate ---- x mph
return rate --- x+11
sum of times = 22.5
550/x + 550/(x+11) = 22.5
times x(x+11)
550x + 6050 + 550x = 22.5x(x+11)
1100x + 6050 = 22.5x^2 + 247.5x
22.5x^2 - 852.5x - 6050 = 0
using the formula
x = 44 or a negative
he went at 44 mph going there, and 55 mph coming back
check:
550/44 + 550/55 = 22.5 , YEAHH
To find the rates at which the man traveled to and from New York, we can set up a system of equations based on the given information.
Let's assume the man's rate of traveling to New York is "x" mph. According to the problem, his rate of returning from New York is 11 mph more, which means it is x + 11 mph.
We are also given that the entire trip required 22.5 hours. The time it takes to travel to New York can be calculated by dividing the distance (550 miles) by the rate (x mph). Similarly, the time it takes to return can be calculated using the distance (550 miles) and the rate (x + 11 mph).
So, our system of equations becomes:
550/x + 550/(x + 11) = 22.5
To solve this equation, we need to find a common denominator and then simplify the equation. To find a common denominator, multiply both sides of the equation by "x(x + 11)":
550(x + 11) + 550x = 22.5x(x + 11)
Now distribute and simplify the equation:
550x + 6050 + 550x = 22.5x^2 + 247.5x
1100x + 6050 = 22.5x^2 + 247.5x
Rearrange the equation into a quadratic equation form:
22.5x^2 + 137.5x - 6050 = 0
Now we can solve this quadratic equation using factoring, completing the square, or the quadratic formula. For simplicity, we'll use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 22.5, b = 137.5, and c = -6050. Plugging these values into the quadratic formula:
x = (-137.5 ± √((137.5)^2 - 4 * 22.5 * -6050)) / (2 * 22.5)
Now we can calculate the values of x by simplifying the equation:
x = (-137.5 ± √(18906.25 + 544500)) / 45
x = (-137.5 ± √(563406.25)) / 45
x = (-137.5 ± 750.075) / 45
This gives us two possible values for x:
x₁ = (-137.5 + 750.075) / 45 = 12.013 mph (approximately)
x₂ = (-137.5 - 750.075) / 45 = -20.457 mph (approximately)
However, the negative value doesn't make sense in this context, so we discard it.
Therefore, the man's rate of traveling to New York was approximately 12.013 mph, and his rate of returning from New York was approximately 23.013 mph.