A cube with edges of length 6\sqrt{3} is inscribed in a sphere with a volume of a\pi. What is the value of a ?

Note: The formula for the volume of a sphere is V=\dfrac{4}{3}\pi r^3.



In the equation above, a, b, and c are constants. If the equation represents a circle with center (5,-2) and radius 7, what is the value of c ?


The product of the complex numbers 5-7i and a+bi, where a and b are constants, is 148. What is the sum of a and b? (Note: i=\sqrt{-1})


In the figure above, quadrilateral PQST is divided into three non-overlapping right triangles. the length of \overline{QR} is 12, the length of \overline{QS} is 13, and the length of \overline{ST} is 4. What is the length of \overline{PT} ?


In the xy-plane, line n passes through the origin and intersects line m at the point (10, 2). Line m is perpendicular to line n. What is the value of the x-intercept of line m ?


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Comments (1)

Two things: Images aren’t loading on a PC in Chrome

Also on number 3, I solved it by foiling 5-7i and a+bi, then the bi and ai elements equal to each other (because they have to equal zero), solving for b and then plugging -7a/5 into the equation 5a-7bi^2 = 48 o get a = 10, which I plugged back into the first equation to get 14. I see that your solution is much more elegant, but my question is this: is the reasoning sound here? Or did I get lucky and this wouldn’t work for other examples? Sorry for the long question!

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