We have, = tan r h α Now, since is similar to we have: ΔAOG ΔCEG, Now, the volume (V) of the cylinder is given by, And, for, we have: second derivative test, the volume of the cylinder is the greatest when ∴By Thus, the height of the cylinder is one-third the height of the cone when the volume of the cylinder is the greatest. Now, the maximum volume of the cylinder can be obtained as: Hence, the given result is proved. Q.19 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h Let be the radius of the cylinder. r Then, volume (V) of the cylinder is given by, Differentiating with respect to time we have: t,
We have, r = h tan α Now, since ΔAOG is similar to ΔCEG, we have: Now, the volume (V) of the cylinder is given by, And, for, we have: ∴By second derivative test, the volume of the cylinder is the greatest when Thus, the height of the cylinder is one-third the height of the cone when the volume of the cylinder is the greatest. Now, the maximum volume of the cylinder can be obtained as: Hence, the given result is proved. Q.19 A cylindrical tank of radius 10 m is being filled with wheat at the rate of 314 cubic mere per hour. Then the depth of the wheat is increasing at the rate of (A) 1 m/h (B) 0.1 m/h (C) 1.1 m/h (D) 0.5 m/h Let r be the radius of the cylinder. Then, volume (V) of the cylinder is given by, Differentiating with respect to time t, we have:
misc_application-diff.docx