"A mirror manufacturer plans to start production of a new series. The new mirrors must have a surface area of 1 square meter and be shaped like a rectangle with a semicircle in the right and left end.
Around the mirror there should be a frame. Production of the curved part
of the frame is twice as expensive per centimeter as the straight part.
How many centimeters should the radius of the half circles be for the frame to be as cheap as possible to manufacture?"
Around the mirror there should be a frame. Production of the curved part
of the frame is twice as expensive per centimeter as the straight part.
How many centimeters should the radius of the half circles be for the frame to be as cheap as possible to manufacture?"
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area of mirror = 1 m² = 10000 cm²
area of both semicircles = πr² cm²
area of rectangle = 10000 - πr² cm²
height of rectangle = 2r cm
width of rectangle = area/height = (10000 - πr²)/(2r) cm
perimeter = curved edges + straight edges = 2πr + (10000 - πr²)/r cm
c = cost per cm of straight edge
f(r) = 2πr(2c) + (10000 - πr²)c/r = 4πrc + 10000c/r - πrc
Set first derivative to zero.
f'(r) = 4πc - 10000c/r² - πc = 3πc - 10000c/r² = 0
3πc = 10000c/r²
r² = 10000/(3π)
r = 100/√(3π) ≅ 32.57cm but there are a lot of numbers here so be sure to check my calcs.
area of both semicircles = πr² cm²
area of rectangle = 10000 - πr² cm²
height of rectangle = 2r cm
width of rectangle = area/height = (10000 - πr²)/(2r) cm
perimeter = curved edges + straight edges = 2πr + (10000 - πr²)/r cm
c = cost per cm of straight edge
f(r) = 2πr(2c) + (10000 - πr²)c/r = 4πrc + 10000c/r - πrc
Set first derivative to zero.
f'(r) = 4πc - 10000c/r² - πc = 3πc - 10000c/r² = 0
3πc = 10000c/r²
r² = 10000/(3π)
r = 100/√(3π) ≅ 32.57cm but there are a lot of numbers here so be sure to check my calcs.