Answer:
a. 9200 in³
b. 4600 in³
c. 20.6 in
Step-by-step explanation:
In all the parts of the question, the balloon is assumed to be in the shape of a sphere. Thus, its volume is given by the formula below.
[tex]\boxed{\text{Volume of sphere}=\frac{4}{3} \pi r^{3} }[/tex]
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a. Diameter= 2(radius)
Radius
= 26 ÷2
= 13 in.
Volume of fully-inflated balloon
[tex]=\frac{4}{3}(\pi )(13^{3} )[/tex]
[tex]=\frac{8788}{3}\pi[/tex]
= 9200 in³ (3 s.f.)
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b. Volume of half-inflated balloon
= [tex]\frac{1}{2}[/tex](volume of fully-inflated balloon)
[tex]=\frac{1}{2} (\frac{8788}{3}\pi )[/tex]
[tex]=\frac{4394}{3}\pi[/tex]
= 4600 in³ (3 s.f.)
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c. Equate the formula with the volume of the half-inflated balloon in terms of [tex]\pi[/tex].
[tex]\frac{4}{3}\pi r^{3} =\frac{4394}{3}\pi[/tex]
Divide both sides by [tex]\bf{\pi}[/tex]:
[tex]\frac{4}{3}r^{3} =\frac{4394}{3}[/tex]
Divide both sides by [tex]\bf{\frac{4}{3}}[/tex]:
[tex]r^3=\frac{4394}{3} \div \frac{4}{3}[/tex]
[tex]r^3= \frac{2197}{2}[/tex]
Cube root both sides:
[tex]r=\sqrt[3]{\frac{2197}{2} }[/tex]
r= 10.318 in. (5 s.f.)
Now that we have found the radius of the half-inflated balloon, we can find its diameter.
Diameter= 2×radius
Diameter of inflated balloon
= 2(10.318)
= 20.6 in. (3 s.f.)
Additional:
To learn more about volume of spheres, do check out the following!
