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The volume of a spherical sculpture is 256 ft³. Rhianna wants to estimate the surface area of the sculpture. To do the estimate, she approximates π using 3 in both the surface area and volume formulas for a sphere.

Using this method, what value does she get for the approximate surface area of the sculpture?

 

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asookles  May 15, 2018
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The volume of a spherical sculpture is 256 ft³. Rhianna wants to estimate the surface area of the sculpture.

To do the estimate, she approximates π using 3 in both the surface area and volume formulas for a sphere.

Using this method, what value does she get for the approximate surface area of the sculpture?

 

Formula:

\(\text{The volume of a sphere of radius $r$ is: $V = \frac{4}{3}\pi r^3$ } \\ \text{The surface area of a sphere of radius $r$ is: $A = 4 \pi r^2$ }\)

 

\(\pi = 3\)

\(\begin{array}{|rcll|} \hline V &=& \frac{4}{3}\cdot 3 r^3 \\ &=& 4r^3 \\\\ A &=& 4 \cdot 3 r^2 \\ &=& 12r^2 \\ \hline \end{array} \)

 

\(r=\ ?\)

\(\begin{array}{|rcll|} \hline V &=& 4r^3 \quad & | \quad V = 256\ ft^3 \\ 256 &=& 4r^3 \\ r^3 &=& \frac{256}{4} \\ r^3 &=& 64 \\ r^3 &=& 4^3 \\ \mathbf{r} &\mathbf{=}& \mathbf{4} \\ \hline \end{array} \)

 

\(A=\ ?\)

\(\begin{array}{|rcll|} \hline \frac{A}{V} &=& \frac{12\cdot r^2}{4r^3} \\ \frac{A}{V} &=& \frac{3}{r} \quad & | \quad r = 4\ ft \\ \frac{A}{V} &=& \frac{3}{4} \\ A &=& \frac{3}{4}V \quad & | \quad V = 256\ ft^3 \\ A &=& \frac{3}{4}\cdot 256 \\ A &=& 3\cdot 64 \\ \mathbf{A} &\mathbf{=}& \mathbf{192\ ft^2} \\ \hline \end{array}\)

 

\(\text{The approximate surface area of the sculpture is $192\ ft^2$ }\)

 

laugh

heureka  May 16, 2018

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