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The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds is
- a)9
- b)10
- c)11
- d)12
Correct answer is option 'A'. Can you explain this answer?
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The surface area of a balloon of spherical shape being inflated increa...
Given:
- The surface area of a balloon of spherical shape increases at a constant rate.
- Initially, the radius of the balloon is 3 units.
- After 5 seconds, the radius becomes 7 units.
To find:
- The radius of the balloon after 9 seconds.
Solution:
Let's first understand the relationship between the radius and the surface area of a sphere.
The surface area of a sphere is given by the formula:
Surface Area = 4πr^2, where r is the radius of the sphere.
Since the surface area of the balloon is increasing at a constant rate, we can write:
Rate of increase of surface area = k, where k is a constant.
Step 1:
We are given that initially the radius of the balloon is 3 units, so the initial surface area (SA1) is:
SA1 = 4π(3^2) = 36π
Step 2:
After 5 seconds, the radius of the balloon becomes 7 units, so the surface area after 5 seconds (SA2) is:
SA2 = 4π(7^2) = 196π
Step 3:
Now, let's find the rate of increase of the surface area.
Rate of increase of surface area = (SA2 - SA1) / (Time2 - Time1)
Rate of increase of surface area = (196π - 36π) / (5 - 0)
Rate of increase of surface area = 160π / 5
Rate of increase of surface area = 32π
Since the rate of increase of the surface area is constant, it will remain the same for any time interval.
Step 4:
Now, let's find the surface area after 9 seconds (SA3):
Rate of increase of surface area = (SA3 - SA2) / (Time3 - Time2)
32π = (SA3 - 196π) / (9 - 5)
32π * 4 = SA3 - 196π
SA3 = 128π + 196π
SA3 = 324π
Step 5:
Finally, let's find the radius of the balloon after 9 seconds.
SA3 = 4π(r^2)
324π = 4π(r^2)
81 = r^2
r = √81
r = 9
Therefore, the radius of the balloon after 9 seconds is 9 units.
Hence, the correct answer is option 'A', 9.
- The surface area of a balloon of spherical shape increases at a constant rate.
- Initially, the radius of the balloon is 3 units.
- After 5 seconds, the radius becomes 7 units.
To find:
- The radius of the balloon after 9 seconds.
Solution:
Let's first understand the relationship between the radius and the surface area of a sphere.
The surface area of a sphere is given by the formula:
Surface Area = 4πr^2, where r is the radius of the sphere.
Since the surface area of the balloon is increasing at a constant rate, we can write:
Rate of increase of surface area = k, where k is a constant.
Step 1:
We are given that initially the radius of the balloon is 3 units, so the initial surface area (SA1) is:
SA1 = 4π(3^2) = 36π
Step 2:
After 5 seconds, the radius of the balloon becomes 7 units, so the surface area after 5 seconds (SA2) is:
SA2 = 4π(7^2) = 196π
Step 3:
Now, let's find the rate of increase of the surface area.
Rate of increase of surface area = (SA2 - SA1) / (Time2 - Time1)
Rate of increase of surface area = (196π - 36π) / (5 - 0)
Rate of increase of surface area = 160π / 5
Rate of increase of surface area = 32π
Since the rate of increase of the surface area is constant, it will remain the same for any time interval.
Step 4:
Now, let's find the surface area after 9 seconds (SA3):
Rate of increase of surface area = (SA3 - SA2) / (Time3 - Time2)
32π = (SA3 - 196π) / (9 - 5)
32π * 4 = SA3 - 196π
SA3 = 128π + 196π
SA3 = 324π
Step 5:
Finally, let's find the radius of the balloon after 9 seconds.
SA3 = 4π(r^2)
324π = 4π(r^2)
81 = r^2
r = √81
r = 9
Therefore, the radius of the balloon after 9 seconds is 9 units.
Hence, the correct answer is option 'A', 9.
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Community Answer
The surface area of a balloon of spherical shape being inflated increa...
S = 4πr2
r dr = k dt
⇒ r2/2 = kt + c
At t = 0, r = 3
9/2 = C
At t = 5,
At t = 9,
So, r = 9
r dr = k dt
⇒ r2/2 = kt + c
At t = 0, r = 3
9/2 = C
At t = 5,
At t = 9,
So, r = 9
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The surface area of a balloon of spherical shape being inflated increases at a constant rate. If initially, the radius of balloon is 3 units and after 5 seconds, it becomes 7 units, then its radius after 9 seconds isa)9b)10c)11d)12Correct answer is option 'A'. Can you explain this answer?
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