Question

Одновременно начато бурение двух скважин со скоростями 2 м/час и 1 м/час, а через час приступили к бурению третьей скважины. Через некоторое время глубина третьей скважины оказалось равной глубине второй, а ещё через 1,5 часа она сравнялась по глубине с первой скважиной. Найти скорость бурения третьей скважины.

Answer 1

Пусть l метров в час - скорость бурения 3 скважины, а t - время, через которое её глубина стала равной глубине второй скважины. Так как последняя равна 1*t=t метров в час, то получаем уравнение l*(t-1)=t. По условию, l*(t-1+1,5)=l*(t+0,5)=2*(t+1,5). Из первого уравнения  находим l=t/(t-1). Подставляя это выражение во второе уравнение, получаем уравнение t(t+0,5)/(t-1)=(t²+0,5*t)/(t-1)=2t+3, или t²+0,5*t=(2t+3)(t-1), или t²+0,5*t=2t²+t-3, или t²+0,5t-3=0, или 2t²+t-6=0. Дискриминант D=1²-4*2*(-6)=49=7². Отсюда t=(-1+7)/4=1,5 часа, а l=t/(t-1)=1,5/0,5=3 метра в час. Ответ: 3 метра в час.

Answer 2

Problem Analysis

We are given that two wells are being drilled simultaneously, one at a rate of 2 m/h and the other at a rate of 1 m/h. After one hour, drilling begins on a third well. We are told that after some time, the depth of the third well is equal to the depth of the second well, and 1.5 hours later, it is equal to the depth of the first well. We need to find the drilling rate of the third well.

Solution

Let's assume that the drilling rate of the third well is x m/h.

After one hour, the first well has drilled 2 m, and the second well has drilled 1 m. The third well has not drilled anything yet.

After some time, the depth of the third well is equal to the depth of the second well. Let's say the time taken for this is t1 hours.

So, the depth of the third well after t1 hours is 1 + x * t1.

1.5 hours later, the depth of the third well is equal to the depth of the first well. Let's say the time taken for this is t2 hours.

So, the depth of the third well after t2 hours is 2 + x * (t1 + 1.5).

Since the depth of the third well is equal to the depth of the second well after t1 hours, we can equate the two expressions:

1 + x * t1 = 2 + x * (t1 + 1.5).

Simplifying the equation, we get:

x * t1 = 1.5 * x + 1.

Simplifying further, we get:

x * t1 - 1.5 * x = 1.

Factoring out x, we get:

x * (t1 - 1.5) = 1.

Finally, solving for x, we get:

x = 1 / (t1 - 1.5).

Therefore, the drilling rate of the third well is 1 / (t1 - 1.5) m/h.

Answer

The drilling rate of the third well is 1 / (t1 - 1.5) m/h.