Question

the volume of this cone is 43,332 cubic centimeters. what is the radius of this cone? round your answer to the nearest hundredth. 46 cm r ≈ \boxed{} centimeters submit

Explanation:

Step1: Recall the volume formula for a cone

The volume \( V \) of a cone is given by the formula \( V=\frac{1}{3}\pi r^{2}h \), where \( r \) is the radius and \( h \) is the height. We know \( V = 43332 \) cubic centimeters and \( h=46 \) centimeters. We need to solve for \( r \).

First, substitute the known values into the formula:
\( 43332=\frac{1}{3}\pi r^{2}(46) \)

Step2: Simplify the equation to solve for \( r^{2} \)

Multiply both sides of the equation by 3 to get rid of the fraction:
\( 43332\times3=\pi r^{2}(46) \)
\( 129996 = 46\pi r^{2} \)

Then, divide both sides by \( 46\pi \) to isolate \( r^{2} \):
\( r^{2}=\frac{129996}{46\pi} \)

Calculate the right - hand side. First, calculate \( 46\pi\approx46\times3.1415926535 = 144.513262061 \)

Then, \( \frac{129996}{144.513262061}\approx900.00 \) (approximate value, more accurately: \( \frac{129996}{46\pi}=\frac{129996}{46\times3.14159265}\approx\frac{129996}{144.513262}\approx900.00 \) (we can also do it step by step: \( 129996\div46 = 2826 \), then \( 2826\div\pi\approx2826\div3.14159265\approx900 \))

Step3: Solve for \( r \)

Take the square root of both sides: \( r=\sqrt{900} = 30 \)? Wait, no, wait, let's recalculate. Wait, maybe I made a mistake in calculation.

Wait, let's go back. The volume formula is \( V=\frac{1}{3}\pi r^{2}h \). So \( r^{2}=\frac{3V}{\pi h} \)

Substitute \( V = 43332 \), \( h = 46 \):

\( r^{2}=\frac{3\times43332}{\pi\times46}=\frac{129996}{46\pi} \)

Calculate \( 3\times43332 = 129996 \), \( 46\times\pi\approx144.513 \)

\( \frac{129996}{144.513}\approx900 \) (exactly, because \( 144.513\times900 = 129961.7 \), which is close to 129996. Wait, maybe my approximation of \( \pi \) is a bit off. Let's use a more accurate value of \( \pi\approx3.1415926535 \)

\( 46\pi=46\times3.1415926535 = 144.513261961 \)

\( 129996\div144.513261961\approx900.15 \) (more accurately, let's do the division: \( 144.513261961\times900 = 129961.9357659 \), \( 129996 - 129961.9357659=34.0642341 \), \( 34.0642341\div144.513261961\approx0.2357 \), so \( r^{2}\approx900 + 0.2357=900.2357 \)? Wait, no, I think I messed up the initial substitution. Wait, the volume of the cone is \( V=\frac{1}{3}\pi r^{2}h \), so solving for \( r^{2} \):

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

So \( 3V = 3\times43332 = 129996 \), \( h = 46 \), so \( \frac{129996}{46\times\pi}=\frac{129996}{46}\times\frac{1}{\pi}=2826\times\frac{1}{\pi}\approx2826\div3.14159265\approx900 \) (because \( 3.14159265\times900 = 2827.433385 \), which is very close to 2826. So there is a slight difference due to rounding. So \( r^{2}\approx900 \), then \( r=\sqrt{900}=30 \)? But let's check with the formula.

If \( r = 30 \), \( h = 46 \), then \( V=\frac{1}{3}\pi\times30^{2}\times46=\frac{1}{3}\pi\times900\times46 = 300\pi\times46=13800\pi\approx13800\times3.14159265 = 43332.31 \), which is very close to 43332. So \( r^{2}=\frac{3V}{h\pi}=\frac{3\times43332}{46\pi}=\frac{129996}{46\pi}\approx\frac{129996}{144.513}\approx900 \), so \( r=\sqrt{900}=30.00 \) (when rounded to the nearest hundredth). Wait, but let's do the calculation more precisely.

\( r^{2}=\frac{3\times43332}{46\times\pi}=\frac{129996}{46\pi} \)

\( 46\pi = 46\times3.141592653589793 = 144.51326206513048 \)

\( 129996\div144.51326206513048 = \frac{129996}{144.51326206513048}\approx900.15 \) (using a calculator for more precision: \( 144.513262065\times900 = 129961.935859 \), \( 129996 - 129961.935859 = 34.064141 \), \( 34.064141\div144.513262065\approx0.2357 \), so \( r^{2}\approx900.2357 \), then \( r=\sqrt{900.2357}\approx30.00 \) (because \( 30^{2}=900 \), \( 30.00^{2}=900 \), \( 30.004^{2}=900.240016 \), which is very close to 900.2357, so \( r\approx30.00 \) when rounded to the nearest hundredth? Wait, but let's check with the volume formula.

If \( r = 30 \), \( h = 46 \), then \( V=\frac{1}{3}\times\pi\times30^{2}\times46=\frac{1}{3}\times\pi\times900\times46 = 300\times46\times\pi=13800\pi\approx13800\times3.14159265 = 43332.31 \), which is very close to the given volume of 43332. So the radius is approximately 30.00 centimeters. Wait, but maybe I made a mistake in the height? Wait, the diagram shows the height as 46 cm. So with the given volume, the radius is approximately 30.00 cm.

Wait, let's do the calculation again:

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

\( 43332=\frac{1}{3}\pi r^{2}(46) \)

Multiply both sides by 3: \( 129996 = 46\pi r^{2} \)

Divide both sides by \( 46\pi \): \( r^{2}=\frac{129996}{46\pi} \)

Calculate \( 46\pi\approx144.51326 \)

\( \frac{129996}{144.51326}\approx900.15 \) (using a calculator: \( 129996\div144.51326\approx900.15 \))

Then \( r=\sqrt{900.15}\approx30.00 \) (because \( 30^{2}=900 \), \( 30.0025^{2}=900.15000625 \), which is very close to 900.15, so when rounded to the nearest hundredth, \( r\approx30.00 \))

Answer:

\( 30.00 \)