Differentials Solution 25

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SOLUTION 25: The cube has an edge length of

$30$ cm and it's frosting is

$0.5$ cm thick. Recall that the volume of a cube of edge length

$x$ is

$V = x^3$

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and choose

$x-values: 30 \rightarrow 30 +(0.5 + 0.5)= 31$
so that

$\Delta x = 31-30 = 1$

The derivative of

$V$ is

$V'= 3 x^2$

$\ \ \ \$ a.) The exact volume of the frosting is the exact change of

$V-$ values and is

$\Delta V = V(31) - V(30)$

$= (31)^3 - (30)^3$

$= 29,791 - 27,000$

$= 2791 \ cm^3$

$\ \ \ \$ b.) An estimate for volume of the frosting is the Differential of

$V$ (since

$\Delta V \approx dV)$ and is

$dV = V'(30) \ \Delta x$

$= 3 (30)^2 \cdot (1)$

$= 2700 \ cm^3$

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