Question

multiple choice 1 point right triangle abc has legs whose lengths are 4 and 6. what is the volume of the three - dimensional object formed by continuously rotating the right triangle around segment ab? 48π 96π 144π 32π clear my selection

Explanation:

Step1: Identify the solid formed

Rotating a right triangle around one of its legs forms a cone. Here, rotating around \( AB \) (length 6), the other leg \( AC \) (length 4) becomes the radius \( r = 4 \), and \( AB \) becomes the height \( h = 6 \) of the cone.

Step2: Recall the volume formula for a cone

The volume \( V \) of a cone is given by \( V=\frac{1}{3}\pi r^{2}h \).

Step3: Substitute the values

Substitute \( r = 4 \) and \( h = 6 \) into the formula:
\( V=\frac{1}{3}\pi(4)^{2}(6) \)
First, calculate \( (4)^{2}=16 \). Then, \( 16\times6 = 96 \). Then, \( \frac{1}{3}\times96 = 32 \). So, \( V = 32\pi \). Wait, no, wait—wait, maybe I mixed up radius and height. Wait, if we rotate around \( AB \), which is length 6, then the leg perpendicular to \( AB \) is \( AC = 4 \), so the radius is \( AC = 4 \), height is \( AB = 6 \). Wait, but let's recalculate: \( \frac{1}{3}\pi r^{2}h=\frac{1}{3}\pi(4)^{2}(6)=\frac{1}{3}\pi\times16\times6=\frac{96}{3}\pi = 32\pi \)? Wait, but the options have 48π, 96π, 144π, 32π. Wait, maybe I rotated around the wrong leg. Wait, maybe rotating around \( AC \) (length 4), then \( AB = 6 \) is the radius. Let's check that. If we rotate around \( AC \) (length 4), then radius \( r = 6 \), height \( h = 4 \). Then volume \( V=\frac{1}{3}\pi(6)^{2}(4)=\frac{1}{3}\pi\times36\times4 = 48\pi \). Ah, that must be it. The problem says "rotating the right triangle around segment \( AB \)"? Wait, no, the problem says: "rotating the right triangle around segment \( AB \)". Wait, \( AB \) is length 6, \( AC \) is length 4, right angle at \( A \). So when rotating around \( AB \), the side \( AC \) (length 4) is the radius, \( AB \) (length 6) is the height. Wait, but then \( V=\frac{1}{3}\pi(4)^2(6)=\frac{1}{3}\pi\times16\times6 = 32\pi \). But the options include 48π. Wait, maybe I misread the rotation axis. Wait, maybe the triangle is rotated around \( AC \)? Wait, the problem says "rotating the right triangle around segment \( AB \)". Let me re-express the triangle: \( A \) is the right angle, \( AC = 4 \), \( AB = 6 \), so \( BC \) is the hypotenuse. Rotating around \( AB \): the radius is \( AC = 4 \), height is \( AB = 6 \). Then volume is \( \frac{1}{3}\pi r^2 h=\frac{1}{3}\pi(4)^2(6)=32\pi \). But the options have 32π as an option. Wait, the options are 48π, 96π, 144π, 32π. So 32π is an option. Wait, but maybe I made a mistake. Wait, let's check again. If we rotate around \( AB \), the radius is \( AC = 4 \), height \( AB = 6 \). So \( V=\frac{1}{3}\pi(4)^2(6)=\frac{1}{3}\pi\times16\times6 = 32\pi \). So the correct answer should be 32π? Wait, but let's check the problem statement again: "Right triangle ABC has legs whose lengths are 4 and 6. What is the volume of the three - dimensional object formed by continuously rotating the right triangle around segment AB?" So legs are 4 and 6, right angle at A, so AB and AC are legs. So rotating around AB (length 6), so AC (length 4) is the radius, AB (length 6) is the height. Then volume is \( \frac{1}{3}\pi r^2 h=\frac{1}{3}\pi(4)^2(6)=32\pi \). So the answer is 32π. Wait, but the options include 32π. So that's the correct one.

Wait, but earlier when I thought of rotating around AC, I got 48π, but the problem says rotating around AB. So the correct volume is 32π.

Answer:

32π (the option with "32π")