Emoji.
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me don't know.
school
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...school!

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6767

An ice sculpture in the form of a sphere is melting. Its spherical shape is maintained as the volume decreases at a constant rate of 

8π8 pi

 cubic meters per hour. At the moment the radius is 5 meters, what is the rate, in square meters per hour, at which the surface area of the sphere is decreasing? 



(Note: For a sphere of radius 

rr

, the surface area is 

A=4πr2cap A equals 4 pi r squared

 and the volume is 

V=43πr3cap V equals four-thirds pi r cubed

.)



  1. Given 
    dVdt=-8πm3/hrthe fraction with numerator d cap V and denominator d t end-fraction equals negative 8 pi space m cubed / h r
     and we need to find 
    dAdtthe fraction with numerator d cap A and denominator d t end-fraction
     when 
    r=5mr equals 5 space m
    .
  2. The volume and surface area formulas are 
    V=43πr3cap V equals four-thirds pi r cubed
     and 
    A=4πr2cap A equals 4 pi r squared
    .
  3. Differentiate both formulas with respect to time (
    tt
    ) using the chain rule: 
    dVdt=4πr2drdtthe fraction with numerator d cap V and denominator d t end-fraction equals 4 pi r squared d r over d t end-fraction
     and 
    dAdt=8πrdrdtthe fraction with numerator d cap A and denominator d t end-fraction equals 8 pi r d r over d t end-fraction
    .
  4. Substitute the given values into the differentiated volume equation to find 
    drdtd r over d t end-fraction
    -8π=4π(5)2drdtnegative 8 pi equals 4 pi open paren 5 close paren squared d r over d t end-fraction
    , which gives 
    drdt=−225m/hrd r over d t end-fraction equals negative 2 over 25 end-fraction space m / h r
    .
  5. Substitute the values of 
    rr
     and 
    drdtd r over d t end-fraction
     into the differentiated surface area equation: 
    dAdt=8π(5)(−225)=−16π5m2/hrthe fraction with numerator d cap A and denominator d t end-fraction equals 8 pi open paren 5 close paren open paren negative 2 over 25 end-fraction close paren equals negative the fraction with numerator 16 pi and denominator 5 end-fraction space m squared / h r


Answer 
The surface area of the sphere is decreasing at a rate of 
16π5the fraction with numerator 16 pi and denominator 5 end-fraction
square meters per hour when the radius is 5 meters.

6767

how many grades did the biggest grade school have

k-12th