When this triangle is rotated about the \(\displaystyle x\)-axis, the resulting solid of revolution is a cone whose base has radius \(\displaystyle r= 6\), and which has height \(\displaystyle h= 5\). Substitute these values into the formula for the volume of a cone:

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

\(\displaystyle = \frac{1}{3} \pi (6)^{2}(5)\)

\(\displaystyle = 36 \cdot 5 \cdot \frac{1}{3} \pi\)

\(\displaystyle =60 \pi\)

The length of one side of the square is one fourth of its perimeter, or \(\displaystyle 36 \div 4 = 9\). The cone inscribed inside this pyramid has the same height. Its base is the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 9, and, as its radius, half this, or \(\displaystyle 9 \div 2 = \frac{9}{2}\).

The volume of a cone, given radius \(\displaystyle r\) and height \(\displaystyle h\), can be calculated using the formula

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

Set \(\displaystyle r = \frac{9}{2}\) and \(\displaystyle h = 10\):

\(\displaystyle V = \frac{1}{3} \pi \left ( \frac{9}{2} \right )^{2} 10\)

\(\displaystyle = \frac{1}{3} \left ( \frac{81}{4} \right ) 10 \pi\)

\(\displaystyle = \frac{135}{2} \pi\)

The length of one side of the square is the square root of the area, or  \(\displaystyle \sqrt{36} = 6\). The cone inscribed inside this pyramid will have as its base the circle inscribed inside the square. This circle will have as its diameter the length of one side of the square, or 6, and, as its radius, half this, or 3.

The volume of a cone, given radius \(\displaystyle r\) and height \(\displaystyle h\), can be calculated using the formula

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

Set \(\displaystyle r = 3\) and \(\displaystyle h = 10\):

\(\displaystyle V = \frac{1}{3} \pi \left (3 \right )^{2} 10\)

\(\displaystyle = \frac{1}{3} \left ( 9 \right ) 10 \pi\)

\(\displaystyle =30 \pi\)

A cone with the same height as a given cylinder and a base the same radius as those of that cylinder has as its volume one-third that of the cylinder. That makes the volume of the cone one-third of 120, or

\(\displaystyle V = \frac{1}{3}(120) = 40\)

The volume of a cone, given radius \(\displaystyle r\) and height \(\displaystyle h\), can be calculated using the formula

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\).

We are given that \(\displaystyle h = 10\), but we are not given the value of \(\displaystyle r\). We are given that \(\displaystyle l = 20\), and since the cone is a right cone, its radius, height, and slant height can be related using the Pythagorean relation

\(\displaystyle r^{2}+ h^{2}= l^{2}\).

Substituting 10 for \(\displaystyle h\) and 20 for \(\displaystyle l\), we can find \(\displaystyle r\):

\(\displaystyle r^{2}+ 10^{2} = 20^{2}\)

\(\displaystyle r^{2}+ 100=400\)

\(\displaystyle r^{2} + 100- 100 =400 - 100\)

\(\displaystyle r^{2} =300\), which is what we need in the formula.

Now substitute in the volume formula:

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

\(\displaystyle \approx \frac{1}{3}(3.1416 )(300)(10)\)

\(\displaystyle \approx 3141.6\)

This rounds to 3,142

The volume of a cylinder with a base of radius \(\displaystyle r\) and with height \(\displaystyle h\) is

\(\displaystyle V = \pi r^{2}h\)

The cone has radius twice that of the cylinder, which is \(\displaystyle 2r\). Its height is 20% greater than, or 120% of, that of the cylinder, which is equal to \(\displaystyle 1.2h\).

The volume of a cone with a base of radius \(\displaystyle r'\) and height \(\displaystyle h'\) is

\(\displaystyle V = \frac{1}{3}\pi r'^{2}h'\)

Set \(\displaystyle r' = 2r\) and \(\displaystyle h'= 1.2h\):

\(\displaystyle V = \frac{1}{3}\pi (2r)^{2} \left (1.2h \right )\)

\(\displaystyle = \frac{1}{3}\pi (2^{2} r^{2} )\left (1.2h \right )\)

\(\displaystyle = \frac{1}{3}\pi (4 r^{2} )\left (1.2h \right )\)

\(\displaystyle = \left (\frac{1}{3} \cdot 4 \cdot 1.2 \right ) \pi r^{2} h\)

\(\displaystyle = 1.6 \pi r^{2} h\)

Substitute \(\displaystyle V\) for \(\displaystyle \pi r^{2}h\):

\(\displaystyle V' =1.6V\).

This means that the volume of the cone is \(\displaystyle 1.6 = 1.6 \times 100\% = 160\%\) times that of  the cylinder—or 60% greater.

The vertices of the triangle are the points of intersection of the three lines. The \(\displaystyle x\)-axis and the \(\displaystyle y\)-axis meet at the origin \(\displaystyle (0,0)\).

The point of intersection of the \(\displaystyle x\)-axis and the graph of \(\displaystyle 2x + y = 8\)—the \(\displaystyle x\)-intercept of the latter—can be found by substituting 0 for \(\displaystyle y\):

\(\displaystyle 2x + y = 8\)

\(\displaystyle 2x + 0= 8\)

\(\displaystyle 2x = 8\)

Divide both sides by 2 to isolate \(\displaystyle x\):

\(\displaystyle 2x \div 2 = 8 \div 2\)

\(\displaystyle x = 4\)

The point of intersection is at \(\displaystyle (4,0)\).

Similarly, the point of intersection of the \(\displaystyle y\)-axis and the graph of \(\displaystyle 2x + y = 8\)—the \(\displaystyle y\)-intercept of the latter—can be found by substituting 0 for \(\displaystyle y\):

\(\displaystyle 2x + y = 8\)

\(\displaystyle 2 (0) + y = 8\)

\(\displaystyle 0 + y= 8\)

\(\displaystyle y = 8\)

The point of intersection is at \(\displaystyle (0,8)\).

The three vertices of the triangle are at the origin, \(\displaystyle (4,0)\), and \(\displaystyle (0,8)\). When this triangle is rotated about the \(\displaystyle y\)-axis, the resulting solid of revolution is a cone whose base has radius \(\displaystyle r= 8\), and which has height \(\displaystyle h= 4\). Substitute these values into the formula for the volume of a cone:

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

\(\displaystyle = \frac{1}{3} \pi (8^{2}) (4)\)

\(\displaystyle = \frac{1}{3}\cdot 64 \cdot 4 \cdot \pi\)

\(\displaystyle = \frac{256}{3} \pi\)

The volume of a cone, given radius \(\displaystyle r\) and height \(\displaystyle h\), can be calculated using the formula

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\).

We are given that \(\displaystyle r = 10\) and \(\displaystyle h= 20\), so we can substitute and calculate:
\(\displaystyle V = \frac{1}{3} \pi (10^{2} )(20)\)

\(\displaystyle \approx \frac{1}{3} \cdot 3.1416 \cdot 100 \cdot 20\)

\(\displaystyle \approx 2094.4\)

To the nearest whole, this is 2,094.

The volume of a cylinder with a base of radius \(\displaystyle r\) and with height \(\displaystyle h\) is

\(\displaystyle V = \pi r^{2}h\)

The cone has height twice that of the cylinder, which is \(\displaystyle 2h\). Its radius is 20% greater than, or 120% of, that of the cylinder, which is equal to \(\displaystyle 1.2r\).

The volume of a cone with a base of radius \(\displaystyle r'\) and height \(\displaystyle h'\) is

\(\displaystyle V = \frac{1}{3}\pi r'^{2}h'\)

Set \(\displaystyle r'=1.2r\) and \(\displaystyle h' = 2h\):

\(\displaystyle V = \frac{1}{3}\pi \left ( 1.2r\right )^{2}(2h)\)

\(\displaystyle = \frac{1}{3}\pi \left ( 1.2^{2} \right )r^{2} (2h)\)

\(\displaystyle = \frac{1}{3} \left ( 1.44 \right ) (2) \cdot \pi r^{2} h\)

\(\displaystyle =0.96 \cdot \pi r^{2} h\)

Substitute \(\displaystyle V\) for \(\displaystyle \pi r^{2}h\):

\(\displaystyle V'= 0.96 V\)

This means that the volume of the cone is \(\displaystyle 0.96= 0.96 \times 100\% = 96\%\) that of the cylinder—or  \(\displaystyle (100 - 96) \% = 4 \%\) less.

The vertices of the triangle are the points of intersection of the three lines. The \(\displaystyle x\)-axis and the \(\displaystyle y\)-axis meet at the origin \(\displaystyle (0,0)\).

The point of intersection of the \(\displaystyle x\)-axis and the graph of \(\displaystyle 2x + y = 8\) - the \(\displaystyle x\)-intercept of the latter - can be found by substituting 0 for \(\displaystyle y\):

\(\displaystyle 2x + y = 8\)

\(\displaystyle 2x + 0= 8\)

\(\displaystyle 2x = 8\)

Divide both sides by 2 to isolate \(\displaystyle x\):

\(\displaystyle 2x \div 2 = 8 \div 2\)

\(\displaystyle x = 4\)

The point of intersection is at \(\displaystyle (4,0)\).

Similarly, The point of intersection of the \(\displaystyle y\)-axis and the graph of \(\displaystyle 2x + y = 8\) - the \(\displaystyle y\)-intercept of the latter - can be found by substituting 0 for \(\displaystyle y\):

\(\displaystyle 2x + y = 8\)

\(\displaystyle 2 (0) + y = 8\)

\(\displaystyle 0 + y= 8\)

\(\displaystyle y = 8\)

The point of intersection is at \(\displaystyle (0,8)\).

The three vertices of the triangle are at the origin, \(\displaystyle (4,0)\), and \(\displaystyle (0,8)\). When this triangle is rotated about the \(\displaystyle y\)-axis, the resulting solid of revolution is a cone whose base has radius \(\displaystyle r= 4\), and which has height \(\displaystyle h= 8\). Substitute these values into the formula for the volume of a cone:

\(\displaystyle V = \frac{1}{3} \pi r^{2}h\)

\(\displaystyle = \frac{1}{3} \pi (4)^{2}(8)\)

\(\displaystyle = \frac{1}{3}\cdot 16 \cdot 8 \cdot \pi\)

\(\displaystyle = \frac{128}{3} \pi\)