The volume of a cube is increasing at a rate of 10cm3min. How fast is the surface area increasing when the length of an edge is 30 cm?

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sirpsta3u · Accepted Answer

Step 1  Write the equation of volume in terms of the length of the cube  Step 2  differentiate the equation in step 1 with respect to time and find the rate at which the length is increasing.  Step 3  Write the equation of surface area in terms of the length of the cube.  Step 4  differentiate the equation in step 3 with respect to time and use the answer found in step 2.  If the side length of the cube is x, then its volume is given by  V=x3  Differentiate both sides with respect to x  dVdt=d(x3)dt  Since the volume is increasing at the rate of 10cm3min,dVdt=10  Use the chain rule in the right-hand side  10=3x3−1⋅dxdt  Solve for dx/dt  dxdt=103x2  Step 2  If the side length of the cube is x, then its surface area is given by  S=6x2  Differentiate both sides with respect to x  dSdt=d(6x2)dt  Use the chain rule in the right-hand side  dSdt=6⋅2x2−1⋅dxdt  Substitute dxdt=103x2  dSdt=12x⋅103x2  dSdt=40x  When the edge length is 30 cm, we have  dSdtx=30=4030=43cm2min

soanooooo40 · Accepted Answer

V=x3  dVdt=3x2dxdt  10=3x2dxdt  dxdt=103x2  SA=6x2  dSAdt=12xdxdt  dxdt=dSAdt/(12x)  d(SA)dt/(12x)=103x2  dSAdt=120x3x2  which when x=30  =120302700=43cmmin

The volume of a cube is increasing at a rate

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