related rates: the base of a pyramid-shaped tank is a square with sides of length 12 meters, and the vertex pyramid is 13 meters above the base. the tank is filled to a depth of 5 meters, and water is flowing into the tank at the rate of 2 cubic meters er second. find the rate of change of the depth of water in the tank.
To find the rate of change of the depth of water in the tank, we can use the concept of related rates.
Step 1: Identify the given information:
- Base of the pyramid-shaped tank is a square with sides of length 12 meters.
- Vertex of the pyramid is 13 meters above the base.
- The tank is filled to a depth of 5 meters.
- Water is flowing into the tank at a rate of 2 cubic meters per second.
Step 2: Define the variables:
- Let's denote the depth of the water in the tank as "h" (in meters).
- The volume of the pyramid-shaped tank is given by V = (1/3) * base area * height.
Step 3: Find the relationship between the variables:
- Since the tank is in the shape of a pyramid, the relationship between the depth (h) and the height (13 meters) is given by similar triangles.
- Using the similar triangles, we can say that (13 - h) / h = (13 - 5) / 12 => (13 - h) / h = 8/12.
Step 4: Differentiate both sides of the equation with respect to time:
- Differentiating both sides with respect to time t, we get (d/dt)((13 - h) / h) = (d/dt)(8/12).
- Let's denote the rate of change of depth as dh/dt.
Step 5: Simplify and solve for dh/dt:
- We can simplify the equation (d/dt)((13 - h) / h) = 8/12 as follows:
- Applying the quotient rule, we get ((d/dt)(13 - h) * h - (13 - h) * (d/dt)(h)) / h^2 = 0.
- Simplifying further, we have ((-dh/dt) * h - (13 - h) * (d/dt)(h)) / h^2 = 0.
- Expanding the equation, we get (-dh/dt * h - 13h + h^2) / h^2 = 0.
- Rearranging the terms, we have -dh/dt * h - 13h + h^2 = 0.
- Simplifying, we get dh/dt = (h^2 - 13h) / h.
Step 6: Substitute the given values and solve for dh/dt:
- We know the depth of water in the tank is 5 meters, so we substitute h = 5 into the equation dh/dt = (h^2 - 13h) / h.
- Substituting h = 5, we get dh/dt = (5^2 - 13 * 5) / 5.
- Simplifying further, we have dh/dt = (25 - 65) / 5 = -40 / 5 = -8 meters per second.
Step 7: Interpret the result:
- The negative sign indicates that the depth of water in the tank is decreasing.
- So, the rate of change of the depth of water in the tank is 8 meters per second in the downward direction.
Therefore, the rate of change of the depth of water in the tank is -8 meters per second.