The radius r, in inches, of a spherical balloon is related to the volume V by r(V)= ∛3V/4π
Air is pumped into the balloon so the volume after t seconds is given by V(t)=16+14t.
a. Find the expression for the composite function r(V(t)).
b. What is the exact time in seconds when the radius reaches 12 inches?
For part a I got r(V(t)) = ∛(3(16+14t)/4π)
but for part b is got 515.9 s and it says it's wrong. I've tried it a few times and keep getting a similar answer. What am I doing wrong?
Thanks
Okay, question is not for you then... No offense meant.
😅It is fine... no offence taken!
well, I get 515.87
But the exact time is 8/7 (144π - 1)
For part a, your expression for the composite function is correct: r(V(t)) = ∛(3(16+14t)/4π)
Now let's move on to part b and find the exact time when the radius reaches 12 inches. We'll substitute r = 12 into the expression r(V(t)) and solve for t.
r(V(t)) = ∛(3(16+14t)/4π)
Substituting r = 12,
12 = ∛(3(16+14t)/4π)
Cubing both sides,
12^3 = (3(16+14t)/4π)
1728 = (3(16+14t)/4π)
Multiplying both sides by 4π,
4π(1728) = 3(16+14t)
Simplifying,
6912π = 48 + 42t
6912π - 48 = 42t
Dividing by 42,
t = (6912π - 48)/42
Now let's calculate that value.
t ≈ 165.14 seconds
Therefore, the exact time when the radius reaches 12 inches is approximately 165.14 seconds, not 515.9 seconds.