The formula A=6V^{\frac{2}{3}} relates the surface area A, in square units, of a cube to the volume V, in cubic units. What is the volume, in cubic inches, of a cube with the surface area 486 \textrm{ in}^{2}?
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Question 3
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What is the solution of \Big( 10^{\frac{x}{6}} \Big) \Big( 10^{\frac{x}{8}} \Big)=10^{10}?
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Question 4
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What is the solution of 9^{x-8}=3^{4x-12} ?
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Question 5
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The diagram below shows a "regular" hexagon. The word "regular" here has a specific geometric meaning; it means that all the side lengths are equal and all the angles are congruent.
The diagram indicates that the side length (remember - all sides are the same) is 2(\sqrt{3})^\frac{x}{12} \textrm{ in}.
The diagram also shows us the length of a line called the "apothem" (the dotted line), which goes from the center of the polygon to the midpoint of a side. The length of the apothem is (\sqrt{3})^{\frac{x}{6}} \textrm{ in}.
The hexagon's total area is 18\sqrt{3} \textrm{ in}^{2}. And it is known that the area of a regular polygon is equal to \frac{1}{2} the product of the perimiter length and the apothem length. That is, if A is the total area, a the length of the apothem, and p the total perimeter length (all the sides together), then:
\large{A=\frac{p\cdot a}{2}=\frac{\big[\textrm{(number of sides)}\cdot \textrm{(length of each side)} \big]\cdot \textrm{(length of the apothem)}}{2}}