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)
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
I used V = (4/3)π r^3 , with r = 12 to get
V = 7238.229...
then
7238.229 = 16 +14t
t = 515.8735...
Just like your answer.
using your r(V(t)) = ∛(3(16+14t)/4π)
12 = ∛(3(16+14t)/4π)
cube both sides
1728 = 3(16+14t)/(4π)
21714.68842 = 3(16+14t)
7238.229.. = 16+14t
7222.229.. = 14t
t = 515.8735... same thing.
mmmhhh?
Yeah, it's strange. Maybe the website is just glitching.
I appreciate the help regardless.
Thank you!!!
the exact time is 8/7 (144π - 1)
For part a, you correctly found the expression for r(V(t)) as:
r(V(t)) = ∛(3(16 + 14t)/4π)
Now let's move on to part b. We want to find the exact time in seconds when the radius reaches 12 inches.
To do this, we need to solve the equation r(V(t)) = 12.
Substituting the expression for r(V(t)) we found earlier, we have:
∛(3(16 + 14t)/4π) = 12
Now, let's solve this equation step by step.
1. Cube both sides of the equation to get rid of the cube root:
(∛(3(16 + 14t)/4π))^3 = 12^3
2. Simplify the equation:
(3(16 + 14t)/4π) = 12^3
3. Multiply both sides by 4π to isolate the expression (16 + 14t):
3(16 + 14t) = 12^3 * 4π
4. Simplify the right side of the equation:
3(16 + 14t) = 1728 * π
5. Distribute the 3:
48 + 42t = 1728 * π
6. Subtract 48 from both sides:
42t = 1728 * π - 48
7. Divide both sides by 42:
t = (1728 * π - 48) / 42
Now, let's evaluate the right side of the equation to find the exact value of t:
t ≈ (1728 * π - 48) / 42 ≈ 270.135
Therefore, the exact time in seconds when the radius reaches 12 inches is approximately 270.135 seconds.
If you've been consistently getting 515.9 seconds, it's possible that you made a calculation error along the way or used an incorrect value for π. Verify your calculations and make sure you use the correct value of π (approximately 3.14159) to obtain the correct answer.