Evaluate . ; S is the part of the cone between the planes and .

Question

Evaluate  .  ; S is the part of the cone  between the planes  and  .

Quizplus · Accepted Answer

We are given two cones, let's call them cone A and cone B. The difference between these two cones is that cone A extends from the top of the solid to the first given plane, while cone B extends from the top of the solid to the second given plane.

We want to find the volume of the part of the solid that is between these two given planes. This is simply the difference between the volume of cone A and cone B.

To find the volume of a cone that extends from the top of a solid to a given plane, we can use the formula for the volume of a cone:
V = (1/3)πr^2h

where r is the radius of the base of the cone and h is the height of the cone.

The radius of the base of cone A is given as 54b9eacf_9f8f_afaf955b15f1_TB5971_11, and the height is given as the distance from the top of the solid (11eaa3e5_54b9_eacd_9f8f_d9cc8f5ad3b0_TB5971_11) to the first given plane (11eaa3e5_54b9_ead0_9f8f_e3f1b387f5ea_TB5971_11). So the volume of cone A is:

V(A) = (1/3)π(54b9eacf_9f8f_afaf955b15f1_TB5971_11)^2(11eaa3e5_54b9_ead0_9f8f_e3f1b387f5ea_TB5971_11 - 11eaa3e5_54b9_eacd_9f8f_d9cc8f5ad3b0_TB5971_11)

Similarly, the radius of the base of cone B is given as 54b9eacf_9f8f_afaf955b15f1_TB5971_11, and the height is given as the distance from the top of the solid to the second given plane (11eaa3e5_54b9_ead1_9f8f_7d2d700f75a5_TB5971_11). So the volume of cone B is:

V(B) = (1/3)π(54b9eacf_9f8f_afaf955b15f1_TB5971_11)^2(11eaa3e5_54b9_ead1_9f8f_7d2d700f75a5_TB5971_11 - 11eaa3e5_54b9_eacd_9f8f_d9cc8f5ad3b0_TB5971_11)

Therefore, the volume of the part of the solid that is between the two given planes is:

V(A) - V(B) = (1/3)π(54b9eacf_9f8f_afaf955b15f1_TB5971_11)^2(11eaa3e5_54b9_ead0_9f8f_e3f1b387f5ea_TB5971_11 - 11eaa3e5_54b9_ead1_9f8f_7d2d700f75a5_TB5971_11)

Simplifying this expression, we get:

V = (1/3)π(54b9eacf_9f8f_afaf955b15f1_TB5971_11)^2(11eaa3e5_54ba_11e2_9f8f_41b7ea3a9fdc_TB5971_11)

which is equal to choice C.