A conical container can hold 120π cubic centimeters of water. The diameter of the base of the container is 12 centimeters. The height of the container is __centimeters. If its diameter and height were both doubled, the container's capacity would be__ times its original capacity.

Hey there! I'm happy to help!

ANSWER #1

To find the volume of a cone, you multiply the base by the height and then divide it by three. So, to find the height, we will do it backwards. First, we will multiply by three.

120π×3=360π

Now, we need to divide by the area of the base. To find the area of the base (a circle) you square the radius and multiply it by pi. The radius is half of the diameter, so since the diameter is 12, the radius is 6.

We square six.

6²=36

We multiply by π.

36π

Now, we can divide our 360π by the area of the base.

360π÷36π=10

Therefore, the height is 10 centimeters.

ANSWER #2

Let's find the volume of this doubled container. Our height will be twenty this time and the diameter will be 24, so the radius will be 12.

First let's find the area.

12²π

144π

Now we multiply by the height.

144π×20=2880π

And then we divide by three.

2880π÷3=960π

Now we divide this by the volume of our original container to see how many times bigger it is.

960π÷120π=8

Therefore, the container's capacity would be 8 times its original capacity.

Have a wonderful day! :D

answer: ratio of volume of cone (with 3 times the height and equal diameter) to volume of cylinder is 2 to 1. the value of π is not used.

step-by-step explanation:

radius is diameter/2 = 4.

area of circle is π r^2.

volume of cylinder is height × area of base = hπr^2 = 3(4^2) π = 48π cubic inches.

not knowing the formula for volume of cone, i calculate it using calculus (which is for calculating things).

volume of cone is integral 0 to h area(y) dy. area(y) is π r(y)^2. r(y) runs r to zero as y runs zero to h, r(0) = r, r(h) = 0.

r(y) = r - r/h y

area(y) = π (r - r/h y)^2 = π (r^2 - 2r^2y/h + r^2y^2/h^2) = π r^2(1-2y/h+y^2/h^2)

antiderivative area(y) = π r^2 (y-y^2/h+y^3/(3h^2))

integral 0 to h = π r^2(h-h^2/h+h^3/(3h^2) - (0)) = π r^2(h - h + h/3) = π r^2(h/3) = π/3 r^2 h

volume of this cone is π/3(4^2)18 cubic inches.

= 6 16 π = 96π cubic inches.

cone has twice the volume, 96/48.

cone height is 6 times the cylinder.

ratio of volume formula: π r^2 cancels.

cone over cylinder ratio is 6h/3 to h, 2 to 1.

the correct option is 2.

step-by-step explanation:

according to the given information triangle abc is a right angle triangle with right angle at a. segment ab is 5 and segment ac is 7. point d is on segment bc and angles adb and adc are right angles.

it is given that triangles abd, cad, and cba are similar.

two triangles are similar if their corresponding sides are proportional.

in triangle cba, using pythagoras theorem

                      )

triangle abd and cba are similar,

                            (2)

triangle cad and cba are similar,

                      )

using (1), (2) and (3), we get

hence proved.

therefore option 2 is correct.