Advanced Quizzes — Quiz 21

Questions in this quiz (10)

1. A rectangular prism has dimensions $$a$$, $$b$$, and $$c$$ units. A cylindrical tunnel of radius $$r$$ is drilled diagonally through the prism so that the axis of the cylinder coincides with the space diagonal of the prism. If the volume removed is exactly $$\frac{1}{8}$$ of the volume of the prism and $$a=b=2c$$, which of the following expressions represents $$r$$ in terms of $$c$$?

   • $$\frac{c}{2\sqrt{\pi}}$$
   • $$\frac{c\sqrt{3}}{4\sqrt{\pi}}$$
   • $$\frac{c\sqrt{6}}{8\sqrt{\pi}}$$
   • $$\frac{c\sqrt{12}}{8\sqrt{\pi}}$$
   • $$\frac{c}{\sqrt{6\pi}}$$

2. The force between two charged particles is modeled by $$F=k\frac{q_1q_2}{d^2}$$, where $$k$$ is a constant, $$q_1$$ and $$q_2$$ are the charges, and $$d$$ is the distance between them. Initially, two particles have charges $$2Q$$ and $$3Q$$ and are separated by a distance $$D$$. In a new system, the first charge is multiplied by $$4$$, the second charge is divided by $$3$$, and the distance between the particles is multiplied by $$\frac{1}{2}$$. What is the ratio of the new force to the original force?

   • $$8:3$$
   • $$16:3$$
   • $$32:3$$
   • $$64:3$$

3. At a university, $$70\%$$ of students participate in at least one of the Debate Club or Science Club. Of all students, $$45\%$$ participate in the Debate Club, $$40\%$$ participate in the Science Club, and $$x\%$$ participate in both clubs. If a student chosen at random from those who participate in at least one of the two clubs is twice as likely to participate in Debate only as in both clubs, what is the value of $$x$$?

   • $$10$$
   • $$15$$
   • $$20$$
   • $$25$$

4. A data set consists of $$200$$ distinct positive integers with mean $$M$$ and standard deviation $$S$$. A new data set is formed by increasing each of the $$50$$ smallest integers by $$8$$ and decreasing each of the $$50$$ largest integers by $$8$$. Which of the following statements must be true?I. The mean of the new data set is equal to $$M$$.II. The standard deviation of the new data set is less than $$S$$.III. The median of the new data set remains unchanged.

   • I only
   • I and II only
   • II and III only
   • I, II, and III

5. The function $$h$$ is defined by $$h(x)=a(x-1)(x-9)$$, where $$a$$ is a nonzero constant. The graph of $$y=h(x)$$ has a minimum value of $$-32$$. What is the value of $$a$$?

   • $$1$$
   • $$2$$
   • $$\frac{1}{2}$$
   • $$-\frac{1}{2}$$

6. A factory produces two products, $$P$$ and $$Q$$. Product $$P$$ requires $$1.2$$ hours of labor and $$4$$ kilograms of material per unit. Product $$Q$$ requires $$0.8$$ hours of labor and $$3$$ kilograms of material per unit. The factory must use at least $$240$$ labor hours and no more than $$900$$ kilograms of material each week. In addition, the number of units of $$Q$$ produced must be at least three times the number of units of $$P$$. If the factory produces exactly $$60$$ units of $$P$$, which of the following inequalities represents all possible values for the number of units of $$Q$$, denoted by $$q$$?

   • $$180\le q\le220$$
   • $$180\le q\le\frac{2200}{3}$$
   • $$180\le q\le220.5$$
   • $$210\le q\le220$$

7. A gourmet bakery prepares two types of pastry boxes. A standard box contains $$4$$ croissants and $$3$$ muffins. A premium box contains $$7$$ croissants and $$5$$ muffins. The bakery has $$250$$ croissants and $$180$$ muffins available. Let $$s$$ represent the number of standard boxes and $$p$$ represent the number of premium boxes. If the bakery must prepare at least $$15$$ premium boxes, what is the maximum possible value of $$s$$?

   • $$28$$
   • $$31$$
   • $$35$$
   • $$36$$

8. A solid object is formed by placing a right circular cone directly on top of a right circular cylinder. The cone and the cylinder share the same circular base. The radius of the base is $$4$$ inches. The height of the cone is twice the height of the cylinder. The total volume of the solid is $$320\pi$$ cubic inches. What is the height of the cylinder, in inches?

   • $$6$$
   • $$8$$
   • $$10$$
   • $$12$$

9. $$\frac{\left(x^{\frac{1}{6}}y^{-\frac{1}{4}}\right)^3\cdot\sqrt[4]{x^2y^5}}{\sqrt{x^{-1}y^3}}$$The expression can be written in the form $$x^ay^b$$, where $$a$$ and $$b$$ are rational numbers. What is the value of $$a+b$$?

   • $$\frac{19}{12}$$
   • $$\frac{11}{6}$$
   • $$\frac{1}{2}$$
   • $$\frac{7}{3}$$

10. The function $$g$$ is defined by $$g(x)=p\sqrt{x-q}+r$$, where $$p$$, $$q$$, and $$r$$ are constants. In the $$xy$$-plane, the graph of $$y=g(x)$$ has its endpoint at $$(-2,5)$$. It is also given that $$g(7)$$ < $$g(2)$$. Which of the following must be true?

    • $$p$$ > $$0$$
    • $$q$$ > $$r$$
    • $$g(14)$$ < $$g(0)$$
    • The function $$g$$ has a minimum value of $$5$$