A spherical balloon expands when it is taken from the cold outdoors to the inside of a warm house. If its surface increases 16.0%, by what percentage does the radius of the balloon change?
Seemed like a straight forward question. I thought the radius would be half of the surface area, but that obviously isn't so as the answer key in the back of the book states that the radius increases by 7.7%. If someone could explain to me how you get to that answer, it would be greatly appreciated.
Seemed like a straight forward question. I thought the radius would be half of the surface area, but that obviously isn't so as the answer key in the back of the book states that the radius increases by 7.7%. If someone could explain to me how you get to that answer, it would be greatly appreciated.
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When things expand:
linear measures expand by a factor
square measures (surface area) expands by the square of the factor
cube measures (volume) expands by the cube of the factor.
SA' = 1.16 * SA
f^2 = 1.16
f = sqrt(1.16)
f = 1.077
So linear measures (such as the radius) expand by 7.7%.
Alternate calculation:
A = 4 pi r^2
A' = 4 pi (r')^2
A' = 1.6A
4 pi (r')^2 = 1.6 4 pi r^2
(r')^2 = 1.6(r^2)
r' = 1.077 r
The new radius is 7.7% longer.
linear measures expand by a factor
square measures (surface area) expands by the square of the factor
cube measures (volume) expands by the cube of the factor.
SA' = 1.16 * SA
f^2 = 1.16
f = sqrt(1.16)
f = 1.077
So linear measures (such as the radius) expand by 7.7%.
Alternate calculation:
A = 4 pi r^2
A' = 4 pi (r')^2
A' = 1.6A
4 pi (r')^2 = 1.6 4 pi r^2
(r')^2 = 1.6(r^2)
r' = 1.077 r
The new radius is 7.7% longer.
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It should be 8.0 % The answer key is wrong
A = 4*π*r²
dA = 8*π*r*dr
dA/A = 8*π*r*dr/4*π*r² = 2*dr/r
The fractional (or percent) area change is twice the fractional radius change.
A = 4*π*r²
dA = 8*π*r*dr
dA/A = 8*π*r*dr/4*π*r² = 2*dr/r
The fractional (or percent) area change is twice the fractional radius change.