The spring runoff from melting snow from a certain region is being accumulated in a reservoir. The reservoir is 450 yards long; and its ross-section is an isosceles triangle 40 yards wide at the top and 20square root(3) yards high. Water is flowing into the reservoir at the rate of 200 cubic yards per hour and, simultaneously, water is leaking out from the reservoir at the rate of 50 cubic yards per hour. Find the rate at which the level of the water is rising when the depth of water is 10squareroot(3) yards.
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To find the rate at which the level of the water is rising when the depth of water is 10√3 yards, we need to determine the rate of change of the volume of water in the reservoir with respect to the height of the water.
Let's break down the problem into smaller steps:
Step 1: Determine the volume of the reservoir.
The cross-section of the reservoir is an isosceles triangle with a width of 40 yards at the top and a height of 20√3 yards. The volume of the reservoir can be calculated using the formula for the volume of a triangular prism:
Volume = (1/2) * base * height * length
In this case, the base is 40 yards, the height is 20√3 yards, and the length is 450 yards. Therefore, the volume of the reservoir is:
Volume = (1/2) * 40 yards * 20√3 yards * 450 yards
Step 2: Calculate the rate of change of the volume of water in the reservoir.
Water is flowing into the reservoir at a rate of 200 cubic yards per hour, and water is leaking out at a rate of 50 cubic yards per hour. The rate of change of the volume of water in the reservoir can be calculated as:
Rate of change = inflow rate - outflow rate
Rate of change = 200 cubic yards per hour - 50 cubic yards per hour
Step 3: Determine the rate at which the water level is rising.
We want to find the rate at which the level of the water is rising when the depth of water is 10√3 yards. The depth of water is directly proportional to the volume of water in the reservoir. Since the volume of water is changing at a certain rate, we can find the rate at which the depth is changing by dividing the rate of change of the volume with respect to time by the cross-sectional area of the reservoir.
Rate of water level rising = Rate of change / Cross-sectional area
Cross-sectional area of the reservoir can be calculated using the formula for the area of an isosceles triangle:
Area = (1/2) * base * height
In this case, the base is 40 yards and the height is the depth of water.
Now you have all the information needed to calculate the rate at which the water level is rising when the depth of water is 10√3 yards. Plug in the values into the formulas mentioned above and compute the final answer.