A funnel is constructed by removing a sector form a circular metal sheet with 7 inch radius. Determine the maximum volume of the funnel constructed in this way if the small amount of volume lost at the tip of the funnel is neglected.
To determine the maximum volume of the funnel constructed by removing a sector from a circular metal sheet, we need to find the angle of the sector that will maximize the volume.
Let's denote the angle of the sector as θ. The remaining central angle of the circular metal sheet, after removing the sector, will be 360° - θ.
The volume of a cone, which represents the funnel, can be calculated using the formula: V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone. In this case, the radius of the base of the cone is 7 inches.
To find the height of the cone, we can use trigonometry. From the diagram, it can be observed that the height of the cone is the radius of the circle (7 inches) multiplied by the sine of half the angle of the sector (θ/2).
Therefore, the height of the cone is h = 7sin(θ/2) inches.
Now, we can substitute the values into the volume formula:
V = (1/3)π(7²)(7sin(θ/2))
Simplifying further:
V = (49/3)πsin(θ/2) cubic inches
Since we want to find the maximum volume, we need to find the value of θ that will maximize the sin(θ/2). The maximum value of sin(θ/2) is 1, which occurs when θ/2 = π/2.
Therefore, θ = 2(π/2) = π radians.
Substituting this value into the volume equation:
V = (49/3)πsin(π/2)
Since sin(π/2) = 1:
V = (49/3)π
Thus, the maximum volume of the funnel, neglecting the small amount of volume lost at the tip, is (49/3)π cubic inches.