Answer: 8:1.
I actually came looking for the answer myself, and decided to show how to determine it is in fact an 8:1 ratio.
Step-by-step explanation:
The wording is actually really important. Notice the phrase, "similar solids"? This is exactly the key piece of information needed. We can also see that:
- Cones have a 2:1 Ratio (a:b)
- We must find the ratio of the volumes compared to slant height.
- Step 2 - Set up some cones
Let's set up two sample cones using the provided ratio 2:1. (A:B)
- Cone B: Radius of 4, height of 12
- Cone A: Radius of 2, height of 6.
We can find the length of the slant height using the pythagorean theorem since the height, radius, and slant height make a right triangle. (See image attached.)
*Pythagorean theorem = [tex]a^2+b^2=c^2[/tex]
A and B are the legs of the triangle, the radius and height. The slant height is c, the hypotenuse.
Cone A: [tex]6^2+2^2=l^2\\36+4=l^2\\40 = l^2\\\sqrt{40} = l\\4\sqrt{10} = l\\[/tex]
Cone B: [tex]12^2+4^2=l^2\\144+16=l^2\\ 160=l^2\\\sqrt{160}=l\\4\sqrt{10} = l[/tex]
Ok cool, we got the slant heights now. By looking at that we can confirm there is a 2:1 ratio happening here. Now, we want to solve for volume.
The equation for volume of a cone is: [tex]\frac{1}{3} \pi r^2h[/tex] where r = radius and h = height. Let's use 3.14 as pi.
- Now just substitute and solve.
Cone A: [tex]\frac{1}{3}*3.14*2^2*6 = V\\\frac{1}{3} *3.14*24\\\frac{1}{3}*75.36\\25.12 = V[/tex]
Cone B: [tex]\frac{1}{3}*3.14*4^2*12 = V\\\frac{1}{3} *3.14*192\\\frac{1}{3}*602.88\\200.96 = V[/tex]
Divide 200.96 by 25.12. You get 8! That means the ratio compared to volume is 8:1.
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- Hope this helped!~ A thank you or brainly would be much appreciated.
Have a great day!
- Astro
