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Math  /  Algebra

QuestionА хотоос В PYY өглөений 8 цагт гарсан машин 64 км/ц хурдтай явсаар тогтсон хугацаанд очих естой байв. Өглөөний 11 цагт жижиг эвдрэл гарч 50 минут зогсон засвар хийв. Цааш явахдаа өөр замаар явсан учир зам нь 31 км-ээр ихэсжээ. Энэ замаар явахдаа тэр 70 km/L70 \mathrm{~km} / \mathrm{L} хурдтай явсан боловч В-д очид дугацаанаасаа 1 цаг 5 минутаар хоцорч очив. A, В хотууд хоорондоо хэдэн км зайтай вэ?

Studdy Solution

STEP 1

1. The car departs from city A to city B at 8:00 AM.
2. The initial speed is 64 km/h.
3. There is a 50-minute stop for repairs at 11:00 AM.
4. After the stop, the route increases by 31 km.
5. The speed on the new route is 70 km/h.
6. The car arrives 1 hour and 5 minutes late.
7. We need to find the distance between cities A and B.

STEP 2

1. Calculate the scheduled travel time.
2. Determine the actual travel time.
3. Set up an equation for the distance.
4. Solve the equation for the distance.

STEP 3

Calculate the scheduled travel time.
Let t t be the scheduled travel time in hours. The car was supposed to reach city B without any delays.

STEP 4

Determine the actual travel time.
The car stopped for 50 minutes, which is 5060=56 \frac{50}{60} = \frac{5}{6} hours. The car was also 1 hour and 5 minutes late, which is 1+560=6560=1312 1 + \frac{5}{60} = \frac{65}{60} = \frac{13}{12} hours late.
The actual travel time is t+56+1312 t + \frac{5}{6} + \frac{13}{12} .

STEP 5

Set up an equation for the distance.
The distance traveled at the initial speed is 64t 64t .
The distance traveled after the stop is 70×(t+56+1312−t) 70 \times (t + \frac{5}{6} + \frac{13}{12} - t) .
The total distance is the sum of these two distances, including the additional 31 km.
64t+31=70×(56+1312) 64t + 31 = 70 \times (\frac{5}{6} + \frac{13}{12})

STEP 6

Simplify the equation.
Calculate the time spent traveling after the stop:
56+1312=1012+1312=2312 \frac{5}{6} + \frac{13}{12} = \frac{10}{12} + \frac{13}{12} = \frac{23}{12}
Now substitute back into the equation:
64t+31=70×2312 64t + 31 = 70 \times \frac{23}{12}

STEP 7

Calculate the right side of the equation:
70×2312=161012=8056 70 \times \frac{23}{12} = \frac{1610}{12} = \frac{805}{6}
Now the equation becomes:
64t+31=8056 64t + 31 = \frac{805}{6}

STEP 8

Solve the equation for the distance.
Subtract 31 from both sides:
64t=8056−31 64t = \frac{805}{6} - 31
Convert 31 to a fraction with a denominator of 6:
31=1866 31 = \frac{186}{6}
So:
64t=8056−1866=6196 64t = \frac{805}{6} - \frac{186}{6} = \frac{619}{6}
Divide both sides by 64:
t=6196×64=619384 t = \frac{619}{6 \times 64} = \frac{619}{384}
Calculate the distance:
64t=6196 64t = \frac{619}{6}
The distance between cities A and B is:
6196 \boxed{\frac{619}{6}}

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