A lady drives 180km from her wine farm to Ceres. On her return (at night) she
drove 30km/h slower. Her return takes one hour longer. At what speed did she
drive home?
daytime rate ---- x km/h
nighttime rate --- x-30 km/h
180/(x-30) - 180/x = 1 , saying the difference in their times is 1 hour
multiply each term by x(x-30)
180x - 180(x-30) = x(x-30)
x^2 -30x - 5400 = 0
solve for x, comes out nice, actually the quadratic factors.
Thanks so much for the solution
Let's calculate the lady's speed while driving from her wine farm to Ceres.
Let's assume her speed while driving from her wine farm to Ceres is "x" km/h.
Therefore, her speed while driving from Ceres back to her wine farm would be "x - 30" km/h (as she was driving 30 km/h slower on her return).
The distance between her wine farm and Ceres is 180 km.
We know that time = distance / speed.
For the first trip from her wine farm to Ceres:
Time taken = 180 km / x km/h = 180/x hours.
For the return trip from Ceres to her wine farm:
Time taken = 180 km / (x - 30) km/h = 180/(x - 30) hours.
According to the information given, her return trip took 1 hour longer than her trip to Ceres.
So, we can set up the equation: 180/x + 1 = 180/(x - 30).
To solve this equation, we can multiply through by x(x - 30) to eliminate the denominators.
180(x - 30) + x(x - 30) = 180x.
180x - 5400 + x^2 - 30x = 180x.
x^2 - 30x - 5400 = 0.
We can solve this quadratic equation to find the value of x.
Factoring the equation, we have:
(x - 90)(x + 60) = 0.
Either x - 90 = 0 or x + 60 = 0.
If x - 90 = 0, then x = 90 km/h.
However, if x + 60 = 0, then x = -60 km/h. Since speed cannot be negative, we can ignore this solution.
Therefore, the lady drove at a speed of 90 km/h from Ceres back to her wine farm.
To solve this problem, we can use the concept of average speed. Average speed is calculated by dividing the total distance traveled by the total time taken.
Let's assume the lady's speed while driving from her wine farm to Ceres is 'x' km/h. Since the distance remains constant, we can say that her average speed for the whole trip is also 'x' km/h.
Now, on her return journey, she drives 30 km/h slower than her speed while driving to Ceres. Therefore, her speed while driving back home can be represented as (x - 30) km/h.
We are given that her return journey takes one hour longer than her journey to Ceres. In terms of time, this can be expressed as:
Return journey time = Journey to Ceres time + 1 hour
To calculate the time taken for each journey, we can use the formula:
time = distance / speed
For the first journey (to Ceres), the time taken is:
180 km / x km/h = 180 / x hours
For the return journey, the time taken is:
180 km / (x - 30) km/h = 180 / (x - 30) hours
Now, according to the given information, the return journey takes one hour longer. So, we can set up the following equation:
180 / (x - 30) = 180 / x + 1
To solve this equation, we can cross-multiply and simplify:
180x = 180(x - 30) + x(x - 30) + x
180x = 180x - 5400 + x^2 - 30x + x
180x = 180x - 5400 + x^2 - 30x + x
5400 = x^2 - 30x + x
5400 = x^2 - 29x
Rearranging the equation, we get:
x^2 - 29x - 5400 = 0
Now, we can solve this quadratic equation using factoring, completing the square, or using the quadratic formula.
After solving the equation, we find two possible solutions for x: x = 200 or x = -27. Since speed cannot be negative, we can ignore the negative solution.
Therefore, the lady's speed while driving home is 200 km/h.