Eudora ran from her home to her secret laboratory at an average speed of 12\text{ km/h}12 km/h12, start text, space, k, m, slash, h, end text. She then took one of her jetpacks and flew to her school at an average speed of 76\text{ km/h}76 km/h76, start text, space, k, m, slash, h, end text. Eudora traveled a total distance of 120120120 kilometers, and the entire trip took 222 hours.
How long did Eudora spend running, and how long did she spend flying using her jetpack?
Eudora ran for
hours and flew for
hours using her jetpack.
Let's denote the time Eudora spent running as "x" hours, and the time she spent flying using her jetpack as "y" hours.
We know that the average speed is equal to the total distance divided by the time taken:
For running:
Average speed = Total distance / Time taken
12 km/h = 120 km / x
For flying:
Average speed = Total distance / Time taken
76 km/h = 120 km / y
We also know that the total time taken for the entire trip is 222 hours:
Time taken = Time spent running + Time spent flying
222 = x + y
Now we can solve these two equations simultaneously to find the values of x and y.
From the first equation, x = 120 km / 12 km/h = 10 hours
Substituting the value of x in the second equation, we have 222 = 10 + y
222 - 10 = y
y = 212 hours
Therefore, Eudora spent 10 hours running and 212 hours flying using her jetpack.
To find the time Eudora spent running and flying, we can first calculate the distance she traveled by running and flying.
Let's assume Eudora spent "x" hours running and "y" hours flying.
The distance Eudora traveled by running can be calculated using the formula: Distance = Speed × Time.
So, the distance Eudora ran is 12 km/h × x hours = 12x km.
The distance Eudora traveled by flying is 76 km/h × y hours = 76y km.
Since the total distance Eudora traveled is 120 km, we can write an equation: 12x + 76y = 120.
We also know that the total time for the trip was 222 hours. So, the time spent running plus the time spent flying equals 222 hours: x + y = 222.
Now, we need to solve these two equations simultaneously to find the values of "x" and "y".
Let's multiply the second equation by 12 to eliminate the variable "x": 12x + 12y = 12 × 222.
12x + 76y = 120
12x + 12y = 12 × 222
By subtracting the second equation from the first equation, we can eliminate the "x" variable:
(12x + 76y) - (12x + 12y) = 120 - (12 × 222)
12x - 12x + 76y - 12y = 120 - 12 × 222
64y = 120 - 12 × 222
64y = 120 - 2664
64y = -2544
y = -2544/64
y = -39.75
Since time cannot be negative, we discard the negative value for "y". Therefore, Eudora did not spend any time flying.
To find the time spent running, substitute the value of "y" into the second equation:
x + y = 222
x - 39.75 = 222
x = 222 + 39.75
x = 261.75
So, Eudora spent 261.75 hours running and 0 hours flying using her jetpack.