given the height, h, and the volume V of a certain cylinder, alex uses the formula r=√v/πh to compute its radius, r, to be 10 meters. a second cylinder has the same volume as the first cylinder, but is 25 times taller. what is the radius of the second cylinder? a). 2/5 meter b). 2 meters c). 50 meters d). 250 meters
Mathematics · High School · Thu Feb 04 2021
Answered on
Let's analyze the problem step by step:
The formula given is �=��ℎ
r=πh
V
, where �
r is the radius, �
V is the volume, and ℎ
h is the height.
For the first cylinder:
- �1=10
- r1
- =10 meters,
- ℎ1=ℎ
- h1
- =h.
Now, for the second cylinder:
- ℎ2=25ℎ
- h2
- =25h (25 times taller),
- �2=�1
- V2
- =V1
- (same volume).
Using the formula �=��ℎ
r=πh
V
, let's compare the two cylinders:
For the first cylinder:
�1=�1�ℎ1
r1
=πh1
V1
For the second cylinder:
�2=�2�ℎ2
r2
=πh2
V2
Since �2=�1
V2
=V1
, we can say:
�2=�1�ℎ2
r2
=πh2
V1
Now, substitute �1=10
r1
=10 meters and ℎ2=25ℎ
h2
=25h:
10=�1�(25ℎ)
10=π(25h)
V1
Now, solve for �1
V1
:
�1=10×�×25×ℎ
V1
=10×π×25×h
�1=250�ℎ
V1
=250πh
Now, substitute this back into the formula for �2
r2
:
�2=250�ℎ�(25ℎ)
r2
=π(25h)
250πh
Simplify:
�2=25025
r2
=25
250
�2=10
r2
=10
So, the radius of the second cylinder is 10 meters.
Therefore, the correct answer is b). 2 meters.