A water tank has the shape of a rectangular prism of base 50 cm2. This tank is being filled at the rate of 12 liters per minutes. Find the rate at which the height of the water in the water tank increases; express your answer in millimeters per second.
12 l/min = 200 ml/s = 200 cm^3/s
since the area of the base is 50cm^2, the height is changing at
200 cm^3/s / 50cm^2 = 4 cm/s = 40 mm/s
12 l/min = 200 ml/s = 200 cm^3/s
since the area of the base is 50cm^2, the height is changing at
200 cm^3/s / 50cm^2 = 4 cm/s = 40 mm/s
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40mm/s
Well, well, well, looks like we have a tank in need of some water! Let's get to it, shall we?
First, let's convert the rate of filling from liters per minute to cubic centimeters per second because we're going to deal with the metric system here. There are 1000 cubic centimeters in a liter and 60 seconds in a minute, so 12 liters per minute is equal to 12,000 cubic centimeters per minute, or 200 cubic centimeters per second.
Now, we know that the base of the tank has an area of 50 cm2, and since we're dealing with a rectangular prism, the volume of the tank is the product of the base area and the height. So, the rate at which the volume of water in the tank increases is 200 cm3/s.
Since we're trying to find the rate at which the height of the water increases, we'll consider the derivative of the volume with respect to time. Let's call the height of the water in the tank h, and the rate at which the height increases dh/dt.
The volume V of the tank at any given time t is given by V = 50h.
Differentiating both sides with respect to time, we get:
dV/dt = 50 * (dh/dt)
We know that dV/dt is 200 cm3/s, so we can plug in those values:
200 = 50 * (dh/dt)
Now we can solve for dh/dt:
dh/dt = 200/50
Calculating this, we find that the rate at which the height of the water in the tank increases is 4 cm/s.
But wait, we need the answer in millimeters per second! So, let's convert that. There are 10 millimeters in a centimeter, so 4 cm/s is equal to 40 mm/s.
Therefore, the rate at which the height of the water in the tank increases is 40 millimeters per second.
To find the rate at which the height of the water in the tank increases, we need to relate the rate of filling the tank in liters per minute to the height of the water.
First, let's convert the rate of filling the tank from liters per minute to cubic centimeters per second since the base area is given in square centimeters.
1 liter = 1000 cubic centimeters (cm³)
1 minute = 60 seconds
So the rate of filling the tank is 12 liters per minute = 12,000 cubic centimeters per minute.
To find the rate at which the height of the water increases, we divide the rate of filling the tank by the base area. This gives us the increase in height per minute.
Rate of height increase = rate of filling the tank / base area
Rate of height increase = 12,000 cm³/min / 50 cm²
Now, let's convert the rate of height increase from centimeters per minute to millimeters per second.
1 centimeter = 10 millimeters
1 minute = 60 seconds
So the rate of height increase is (12,000 cm³/min / 50 cm²) * (1 cm / 10 mm) * (1 min / 60 sec) = 4 mm/sec.
Therefore, the rate at which the height of the water in the tank increases is 4 millimeters per second.