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# SAT Direct and Inverse Variation

If $$x$$ and $$y$$ are positive numbers and $$x$$ is directly proportional to $$\frac{1}{y}$$, which of the following is directly proportional to $$\frac{1}{x^{7}}$$?

(A) $$\ \ \frac{1}{y^{7}}$$
(B) $$\ \ \frac{1}{y}$$
(C) $$\ \ y$$
(D) $$\ \ y^{7}$$
(E) $$\ \ y^{8}$$

$$B$$ varies directly with the square of $$n$$ and the cube of $$m$$ and inversely with $$p$$. What is the value of $$B$$ if $$m=7, n=12$$, $$p=84$$, and the constant of proportionality, $$k,$$ is $$k=\frac{1}{21}$$?

(A) $$\ \ \frac{1}{12348}$$

(B) $$\ \ \frac{1}{21}$$

(C) $$\ \ \frac{12}{21}$$

(D) $$\ \ 28$$

(E) $$\ \ 588$$

In which of the following equations is $$y$$ directly proportional to $$x$$?

(A) $$\ \ y=\frac{1}{5}\cdot x^{-2}$$

(B) $$\ \ y=\frac{5}{x}$$

(C) $$\ \ y=\frac{1}{5}\cdot x$$

(D) $$\ \ y=\frac{1}{5}\cdot x^{2}$$

(E) $$\ \ y=5+2x$$

The kinetic energy of an object, $$KE$$, is directly proportional to its mass, $$m$$, and to the square of its velocity, $$v$$. If the mass is divided by $$2$$ and the velocity is multiplied by $$11$$, and if $$k$$ is a proportionality constant, which of the following does NOT represent the object's new kinetic energy, $$KE_{new}$$?

(A) $$\ \ KE_{new}=\frac{121k}{2}\cdot KE_{old}$$

(B) $$\ \ KE_{new}=\frac{121 \cdot k}{2} \cdot m \cdot v^{2}$$

(C) $$\ \ KE_{new}=k\cdot \frac{m}{2} \cdot (v \cdot 11)^{2}$$

(D) $$\ \ KE_{new}=\frac{k\cdot 11}{2} \cdot m \cdot v^{2}$$

(E) $$\ \ KE_{new}= k\cdot 11^{2} \cdot \frac{1}{2}\cdot m \cdot v^{2}$$

$$y$$ varies directly with the square of $$x,$$ and $$y=196$$ when $$x=7$$. Given $$x<0$$, what is $$x$$ when $$y=1764$$?

(A) $$\ \ -63$$
(B) $$\ \ -21$$
(C) $$\ \ 0$$
(D) $$\ \ 21$$
(E) $$\ \ 63$$

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