A cone shaped paper drinking cup is to be made from a circular piece of paper of radius 3 inches by cutting out a sector of the circle and gluing the straight edges together. Find the angle of the cut that gives the cup with the largest capacity.
To find the angle of the cut that gives the cup with the largest capacity, we need to maximize the volume of the cone.
The volume of a cone is given by the formula V = (1/3)πr^2h, where r is the radius of the circular base and h is the height of the cone.
In this case, the radius of the circular base is given as 3 inches. Let's assume that the angle of the cut is θ (in radians), and it creates the height h and slant height s of the cone.
Since the slant height s is the hypotenuse of a right triangle formed by the radius r and height h, we can use trigonometry to find the relationship between s, r, and θ. By drawing the cone's lateral face, we see that sin(θ/2) = r/s, which gives us s = r/sin(θ/2).
Now, we can express the height h in terms of s and θ. Considering the right triangle formed by the height h, the radius r, and the slant height s, we have sin(θ/2) = h/s, which implies h = s⋅sin(θ/2).
Substituting the value of s from the earlier expression, h = (r/sin(θ/2))⋅sin(θ/2) = r.
So, the height of the cone h is simply equal to the radius r.
Now, we can substitute the values of r and h in the formula for the volume of the cone:
V(θ) = (1/3)πr^2h = (1/3)πr^2r = (1/3)πr^3.
Since r is fixed at 3 inches, we need to maximize V(θ), which simplifies to V(θ) = (1/3)π(3^3).
Now, to find the angle of the cut θ that maximizes the volume, we can differentiate V(θ) with respect to θ and set it equal to zero to find the critical point.
dV/dθ = 0.
Differentiating V(θ) = (1/3)π(3^3) with respect to θ gives:
dV/dθ = 0.
Since dV/dθ = 0, we can solve for θ:
0 = (1/3)π(3^3).
After simplifying, we get:
0 = 9π.
Since this equation has no solution, we conclude that there is no critical point with a maximum volume. Therefore, the cone's cup with the largest capacity is obtained by using the full circular piece of paper and making no cut (θ = 2π radians or 360 degrees).