1. A man completing a 40km trip finds that by travelling one more km per hour, he could have made the journey in 2 hours less time. At what speed did he actually travel?
2. One pipe alone can fill a tub in 6 minutes less than it takes a second pipe alo e to fill the same tub. Both pipes together can fill tne tub in 4 minutes. How long does it take each pipe to fill tne tub?
1.
let his actual speed by x km/hr
time to do his 40 km trip = 40/x
let his supposed speed be x+1 km/h
time taken = 40/(x+1)
40/x - 40(x+1) = 2
times x(x+1) , the LCD
40(x+1) - 40x = 2x^2 + 2x
2x^2 + 2x - 40 = 0
x^2 + x - 20 = 0
(x+5)(x-4) = 0
x = 4 or a negative
he went at 4 km/h
2. time to fill tub by slow pipe = x min
rate of that pipe = 1/x
time to fill tub by other pipe = x-6
rate of other pipe = 1/(x-6)
combined rate = 1/x + 1/(x-6)
= (x-6 + x)/(x(x-6))
= (2x - 6)/(x^2 - 6x)
but 1 / (2x - 6)/( (x^2 - 6x) ) = 4
(x^2 - 6x)/(2x - 6) = 4
x^2 - 6x = 8x - 24
x^2 - 14x + 24 = 0
check my arithmetic, and solve this quadratic
To find the answers to these questions, we can use algebraic equations and solve for the variables.
Let's start with the first question:
1. A man completing a 40 km trip finds that by traveling one more km per hour, he could have made the journey in 2 hours less time. At what speed did he actually travel?
Let's assume the man's actual speed to be x km/h. Then, according to the given information, if he traveled one more km per hour, his speed would be (x + 1) km/h. We know that by doing so, he could have made the journey in 2 hours less time.
We can set up the equation based on the distance and time:
40 / (x + 1) = 40 / x - 2
To solve this equation, we can cross multiply and simplify:
40x = 40(x + 1) - 2(x + 1)
40x = 40x + 40 - 2x - 2
40x - 40x = -2 + 40 - 2x
0 = 38 - 2x
2x = 38
x = 19
Therefore, the man actually traveled at a speed of 19 km/h.
Moving on to the second question:
2. One pipe alone can fill a tub in 6 minutes less than it takes a second pipe alone to fill the same tub. Both pipes together can fill the tub in 4 minutes. How long does it take each pipe to fill the tub?
Let's assume the time taken by the first pipe alone to be x minutes. Then, based on the given information, the time taken by the second pipe alone would be (x + 6) minutes. Together, both pipes can fill the tub in 4 minutes.
We can set up the equation based on the rate at which each pipe fills the tub:
1/x + 1/(x + 6) = 1/4
To solve this equation, we can find a common denominator and simplify:
(x + 6 + x) / (x(x + 6)) = 1/4
(2x + 6) / (x^2 + 6x) = 1/4
Cross multiplying and simplifying further:
4(2x + 6) = x^2 + 6x
8x + 24 = x^2 + 6x
0 = x^2 - 2x - 24
0 = (x - 6)(x + 4)
This gives us two possible solutions: x = 6 or x = -4.
Since time cannot be negative, we discard x = -4.
Therefore, it takes the first pipe alone 6 minutes to fill the tub, and it takes the second pipe alone (6 + 6) = 12 minutes to fill the tub.