Advanced Quizzes — Quiz 14

Questions in this quiz (10)

1. A right circular cone is inscribed in a sphere with radius $$9$$ units. The height of the cone is $$\frac{14}{9}$$ times the radius of the sphere. What is the volume of the cone?

   • $$324\pi$$
   • $$378\pi$$
   • $$\frac{784}{3}\pi$$
   • $$486\pi$$

2. The lengths of the sides of a triangle are given by the expressions $$4x-7$$, $$2x+5$$, and $$7x-20$$. For these expressions to form a valid triangle, each side length must be positive, and the sum of the lengths of any two sides must be greater than the length of the third side. Which of the following inequalities represents all possible values of $$x$$?

   • $$\frac{20}{7}$$ < $$x$$ < $$32$$
   • $$5$$ < $$x$$ < $$32$$
   • $$\frac{32}{9}$$ < $$x$$ < $$18$$
   • $$9$$ < $$2x$$ < $$64$$

3. The expression shown is equivalent to $$\frac{3}{4x+1}-\frac{2x-5}{3}$$ and can be written in the form $$\frac{ax^2+bx+c}{12x+3}$$. What is the value of $$a+b+c$$?

   • 24

4. A sphere is perfectly inscribed within a cube. The surface area of the cube is $$864$$ square units. What is the volume, in cubic units, of the sphere?

   • $$288\pi$$
   • $$324\pi$$
   • $$432\pi$$
   • $$500\pi$$

5. A data set $$D$$ consists of $$n$$ positive integers. The mean of the data set is $$M$$, the median is $$N$$, and the range is $$R$$. A new data set $$D'$$ is formed by removing both the largest and smallest integers from $$D$$. Assume $$n\ge4$$. Which of the following statements must be true?I. The range of $$D'$$ is less than $$R$$.II. The mean of $$D'$$ is always equal to $$M$$.III. The median of $$D'$$ is always equal to $$N$$.

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

6. A polling agency conducted a survey of a random sample of $$2,500$$ registered voters in a city. The survey found that $$54\%$$ of the sampled voters indicated they would vote for Candidate Z, with an associated margin of error of $$3\%$$. The city has a total of $$420,000$$ registered voters. Based on these results, which of the following is a plausible value for the number of registered voters in the city who would vote for Candidate Z?

   • 201,600
   • 214,200
   • 240,200
   • 252,000

7. A scientist is monitoring the spread of a bacterial culture in a laboratory. Let $$P(t)$$ represent the population of bacteria after $$t$$ hours. Initially, the population grows slowly, but over time the rate of growth increases proportionally to the current population size. Which of the following types of functions would best model the bacterial population over time?

   • Linear
   • Quadratic
   • Exponential
   • Logarithmic

8. Consider the system of equations: $$\frac{1}{x-1}+\frac{1}{y-1}=\frac{1}{2}$$ $$x+y=8$$ How many real solutions $$(x,y)$$ does this system have?

   • 0
   • 1
   • 2
   • Infinitely many

9. A factory produces two types of devices, Type A and Type B. The ratio of the number of Type A devices produced to Type B devices produced in one day is $$4:7$$. Each Type A device requires $$3$$ hours of labor, and each Type B device requires $$5$$ hours of labor. If the total labor available per day is $$1,316$$ hours, what is the total number of devices produced in one day?

   • 176
   • 224
   • 308
   • 352

10. In right triangle $$\triangle ABC$$, angle $$C$$ is the right angle. If $$\tan(A)=\frac{5}{12}$$ and the length of side $$BC$$ is $$20$$, what is the length of the altitude from vertex $$C$$ to hypotenuse $$AB$$?

    • $$\frac{120}{13}$$
    • $$\frac{240}{13}$$
    • $$\frac{300}{13}$$
    • $$\frac{480}{13}$$