Respuesta :
Answer:
To find the new diameter of the balloon after half of the air is let out, we can use the concept of proportionality.
The volume of a sphere is proportional to the cube of its diameter. So, if the volume of the balloon changes, the diameter will change accordingly.
Initially, the diameter of the balloon is 14 inches. When half of the air is let out, the volume of the balloon is halved.
Let
�
1
V
1
be the initial volume of the balloon and
�
2
V
2
be the volume of the balloon after half of the air is let out.
We know that
�
=
4
3
�
�
3
V=
3
4
πr
3
, where
�
r is the radius of the sphere.
Initially, with a diameter of 14 inches, the radius
�
1
=
14
2
=
7
r
1
=
2
14
=7 inches.
So,
�
1
=
4
3
�
(
7
)
3
=
4
3
�
(
343
)
=
1372
3
�
V
1
=
3
4
π(7)
3
=
3
4
π(343)=
3
1372
π.
When half of the air is let out, the new volume
�
2
V
2
becomes
1372
6
�
6
1372
π.
Now, let's find the new diameter
�
2
d
2
.
Using the formula for the volume of a sphere,
�
2
=
4
3
�
(
�
2
)
3
V
2
=
3
4
π(r
2
)
3
.
Solving for
�
2
r
2
, we get
�
2
=
3
�
2
4
�
3
r
2
=
3
4π
3V
2
.
Substituting the value of
�
2
V
2
, we get
�
2
=
3
⋅
1372
6
�
4
�
3
r
2
=
3
4π
3⋅
6
1372
π
.
�
2
=
3
⋅
1372
24
3
r
2
=
3
24
3⋅1372
.
�
2
=
1372
8
3
r
2
=
3
8
1372
.
�
2
=
171.5
3
r
2
=
3
171.5
.
�
2
≈
5.9
r
2
≈5.9 inches.
Finally, the new diameter
�
2
d
2
is
2
×
�
2
2×r
2
, which is approximately
2
×
5.9
2×5.9 inches.
�
2
≈
11.8
d
2
≈11.8 inches.
So, after half of the air is let out, the new diameter of the balloon is approximately 11.8 inches.
Step-by-step explanation:
