Extra Math Practice #196-200

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 $V=\dfrac{4}{3}\pi r^3$ .

$x^2+y^2+ax+by=c$

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 $c$ ?

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 $m$ ?

In the $xy$ -plane, $(a,b)$ and $(c,d)$ are the points of intersection between the graphs of $y=x+3$ and $y=x^2-3$ . What is the value of $a+b+c+d$ ?

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