Final answer:
The rate at which the level in the pot is rising when the coffee in the cone is 5 inches deep is 72 / pi inches per minute.
Step-by-step explanation:
To find the rate at which the level in the pot is rising, we need to find the rate at which the volume of coffee in the cone is changing. We can use similar triangles to relate the depth of the coffee in the cone to the level in the pot. Let's define the height of the coffee in the cone as 'h' and the corresponding height in the pot as 'x'.
Since the filter and the pot have the same diameter, the ratio of their heights 'h/x' is equal to the ratio of their volumes. The volume of the cone is given by (1/3) * pi * r^2 * h, and the volume of the cylinder is given by pi * r^2 * x. Since the diameter of both the cone and the pot is 6 inches (which means the radius is 3 inches), the volumes can be expressed as (1/3) * pi * (3^2) * h and pi * (3^2) * x, respectively.
Given that the volume of the coffee is decreasing at a rate of 8 cubic inches per minute, we have -(dV/dt) = 8. This rate of change can be found by differentiating the volume equation with respect to time 't': -(1/3) * pi * (3^2) * dh/dt = 8. Rearranging the equation to solve for dh/dt gives us -dh/dt = (8 * 3 * 3) / (1/3 * pi).
Now, we need to find the value for 'dh/dt' when the depth of the coffee in the cone is 5 inches. Using the derived equation, we have -dh/dt = (8 * 3 * 3) / (1/3 * pi) = 72 / pi. Therefore, the rate at which the level in the pot is rising when the coffee in the cone is 5 inches deep is 72 / pi inches per minute.