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## Quiz #105

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Question 1 |

**The circumference of a right circular cylinder is half its height. The radius of the cylinder is x. What is the volume of the cylinder in terms of x?**

$2 \pi x^3$ | |

$3 \pi x^3$ | |

$3$$\pi$$^2$$x^3$ | |

$4$$\pi$$^2$$x^3$ |

Question 1 Explanation:

The correct answer is (D). Begin by noting the relevant, necessary formulas:

Circumference: $C=2 \pi r$

Volume: $V = \pi r^2 h$

Note that radius is equal to

$C$ $=$ $2 \pi r$ $=$ $\frac{1}{2} h$, or $h$ $=$ $4 \pi r$

Substitute this expression for

$V$ $=$ $\pi x^2 (4 \pi x)$ $=$ $4$$\pi$$^2$$x^2$

Circumference: $C=2 \pi r$

Volume: $V = \pi r^2 h$

Note that radius is equal to

*x*and the circumference is equal to half the height, or:$C$ $=$ $2 \pi r$ $=$ $\frac{1}{2} h$, or $h$ $=$ $4 \pi r$

Substitute this expression for

*h*into the volume formula:$V$ $=$ $\pi x^2 (4 \pi x)$ $=$ $4$$\pi$$^2$$x^2$

Question 2 |

**What is the value of**

*a*+*b*?w − x − xy + z | |

360 − x + y + w + z | |

180 − y + z − w | |

w + x + y + z − 360 |

Question 2 Explanation:

The correct answer is (D). Recall that angles forming a straight line and the 3 interior angles of a triangle each sum to 180°. Using these relationships and substituting expressions for the unlabeled angles enables us to then solve the equations in terms of

For the top triangle:

360 +

For the bottom triangle:

360 +

Adding the 2 equations together:

*a*and*b*. We can then combine these equations to solve for*a*+*b*.For the top triangle:

*a*+ (180 −*w*) + (180 −*x*) = 180°360 +

*a*−*w*−*x*= 180*a*= −180 +*w*+*x*For the bottom triangle:

*b*+ (180 −*y*) + (180 −*z*) = 180°360 +

*b*−*y*−*z*= 180*b*= −180 +*y*+*z*Adding the 2 equations together:

*a*+*b*= (−180 +*w*+*x*) + (−180 +*y*+*z*)*a*+*b*= −360 +*w*+*x*+*y*+*z*Question 3 |

**A passenger ship left Southampton, England for the Moroccan coast. The ship travelled the first 230 miles at an average speed of 20 knots, then increased its speed for the next 345 miles to 30 knots. It travelled the remaining 598 miles at an average speed of 40 knots. What was the ship’s approximate average speed in miles per hour? (1 knot = 1.15 miles per hour)**

36 | |

38 | |

40 | |

42 |

Question 3 Explanation:

The correct answer is (A). In order to calculate the average speed of the trip, it is first necessary to solve for the total number of miles traveled and the total length of time traveled. Because the question asks for the average speed in miles per hour (and not knots), it is also necessary to convert all knots into miles per hour (mph).

Use the provided unit conversion to set up a general expression relating knots to mph:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{y knots}{x miles}$

where y = 20, 30, and 40:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{20 knots}{x miles}$

Solving for x:

$x$ $mph$ $=$ $20$

Notice that knots cancel from the numerator and denominator leaving units of mph, which is what we are looking for; this verifies that we are on the right track toward the solution. Repeat this calculation for the other knot values to find: 30 knots = 34.5 mph; 40 knots = 46 mph.

Using the calculated mph values, we can now solve for the length of time each portion of the trip took. Recall that a total distance travelled is equal to a rate, or speed of travel, multiplied with a length of time; or: Distance = Rate * Time. Dimensional analysis is again useful to verify that the equation results in the appropriate units.

Use the distances and corresponding rates to solve for each time:

$230$

$230$

Repeat this calculation for the other times: 345 miles takes 10 hours, and 598 miles takes 13 hours.

Use the calculated values to divide the total distance travelled by the total length of time travelled to determine the overall average rate of travel:

$=$ $\frac{1173 miles}{33 hours}$

$≈$ $35.55 \frac{miles}{hour}$

Use the provided unit conversion to set up a general expression relating knots to mph:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{y knots}{x miles}$

where y = 20, 30, and 40:

$\frac{1 knot}{1.15 miles}$ $=$ $\frac{20 knots}{x miles}$

Solving for x:

$x$ $mph$ $=$ $20$

**knots**$*$ $\frac{1.15 mph}{1 knot}$ $=$ $23$ $mph$Notice that knots cancel from the numerator and denominator leaving units of mph, which is what we are looking for; this verifies that we are on the right track toward the solution. Repeat this calculation for the other knot values to find: 30 knots = 34.5 mph; 40 knots = 46 mph.

Using the calculated mph values, we can now solve for the length of time each portion of the trip took. Recall that a total distance travelled is equal to a rate, or speed of travel, multiplied with a length of time; or: Distance = Rate * Time. Dimensional analysis is again useful to verify that the equation results in the appropriate units.

Use the distances and corresponding rates to solve for each time:

$230$

**miles**$=$ $\frac{23 miles}{hours}$ $*$**Time**$230$

**miles**$*$ $\frac{hours}{23 miles}$ $=$ $10$ $hours$Repeat this calculation for the other times: 345 miles takes 10 hours, and 598 miles takes 13 hours.

Use the calculated values to divide the total distance travelled by the total length of time travelled to determine the overall average rate of travel:

**Average speed**$=$ $\frac{230 + 345 + 598 miles}{10 + 10 + 13 hours}$$=$ $\frac{1173 miles}{33 hours}$

$≈$ $35.55 \frac{miles}{hour}$

**, which is approximately 36 mph**Question 4 |

**At a certain lab, the ratio of scientists to engineers is 5:1. If 75 new team members are hired in the ratio of two engineers per scientist, the new ratio of scientists to engineers would be approximately 2:3. Approximately how many scientists currently work at the lab?**

5 | |

10 | |

15 | |

20 |

Question 4 Explanation:

The correct answer is (B). Currently our ratio is

$\frac{(E + 50)}{(S + 25)}$ $=$ $\frac{3}{2}$

Substitute 5

$\frac{(E + 50)}{(5E + 25)}$ $=$ $\frac{3}{2}$

2(

2

100 = 15

25 = 15

$\frac{25}{13}$ =

1.9 =

Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists.

*S/E*= 5/1, or 5*E*=*S*. Out of the 75 hires, for every 3 hires: two will be engineers and one will be a scientist, so that will add 50 engineers and 25 scientists. We’re told that adding these people to the mix creates a new ratio:$\frac{(E + 50)}{(S + 25)}$ $=$ $\frac{3}{2}$

Substitute 5

*E*for*S*, and solve:$\frac{(E + 50)}{(5E + 25)}$ $=$ $\frac{3}{2}$

2(

*E*+ 50) = 3(5*E*+ 25)2

*E*+ 100 = 15*E*+ 75100 = 15

*E*+ 7525 = 15

*E*$\frac{25}{13}$ =

*E*1.9 =

*E*(or approximately 2)Since there are 5 scientists for every engineer currently, then there are approximately 10 scientists.

Question 5 |

**In the**

*xy*-coordinate plane, a circle with center (−4, 0) is tangent to the line*y*= −*x*. What is the circumference of the circle?$4\pi$ | |

$2\pi \sqrt{2}$ | |

$4\pi$ | |

$4\pi \sqrt{2}$ |

Question 5 Explanation:

The correct answer is (D). Start by drawing a diagram to better visualize the problem.

The line y = −x makes a 45 degree angle with each axis in the second quadrant. Connect the center of the circle to the point of tangency on y = −x. The radius of a circle is perpendicular to its point of tangency.

We can draw a 45-45-90 triangle using the x-axis and y = −x, and use our knowledge of right triangle ratios to find the radius (or hypotenuse of the triangle) is 2√2. Recall that 45°, 45°, 90° right triangles share a side: side: hypotenuse ratio of x: x: x√2.

The line y = −x makes a 45 degree angle with each axis in the second quadrant. Connect the center of the circle to the point of tangency on y = −x. The radius of a circle is perpendicular to its point of tangency.

We can draw a 45-45-90 triangle using the x-axis and y = −x, and use our knowledge of right triangle ratios to find the radius (or hypotenuse of the triangle) is 2√2. Recall that 45°, 45°, 90° right triangles share a side: side: hypotenuse ratio of x: x: x√2.

**Circumference**$=$ $2 \pi r$ $=$ $2 \pi * 2 \sqrt{2}$ $=$ $4\pi \sqrt{2}$Question 6 |

**Under which conditions is the expression**$\frac{ab}{a-b}$

**always less than zero?**

$a$ $<$ $b$ $<$ $0$ | |

$0$ $<$ $b$ $<$ $a$ | |

$a$ $<$ $0$ $<$ $b$ | |

$b$ $<$ $a$ $<$ $0$ |

Question 6 Explanation:

The correct answer is (A). If the expression $\frac{ab}{a-b}$ $<$ $0$, then $ab$ $<$ $0$ or $a$ $-$ $b$ $<$ $0$ → $a$ $<$ $b.$

The only answer that guarantees a negative answer is $a$ $<$ $b$ $<$ $0.$

Consider $a$ $=$ $-2,$ $b$ $=$ $-1;$

$\frac{(-2)*(-1)}{-2-(-1)}$ $=$ $\frac{2}{-1}$ $=$ $-2.$

On the other hand, if $a$ $=$ $-1$ and $b$ $=$ $1$, then:

$\frac{(-1)*(1)}{-1-1}$ $=$ $\frac{-1}{-2}$ $=$ $\frac{1}{2}$

The only answer that guarantees a negative answer is $a$ $<$ $b$ $<$ $0.$

Consider $a$ $=$ $-2,$ $b$ $=$ $-1;$

$\frac{(-2)*(-1)}{-2-(-1)}$ $=$ $\frac{2}{-1}$ $=$ $-2.$

On the other hand, if $a$ $=$ $-1$ and $b$ $=$ $1$, then:

$\frac{(-1)*(1)}{-1-1}$ $=$ $\frac{-1}{-2}$ $=$ $\frac{1}{2}$

Question 7 |

**In a recent survey of two popular best-selling books, two-fifths of the 2,200 polled said they did not enjoy the second book, but did enjoy the first book. Of those, 40% were adults over 18. If three-eighths of those surveyed were adults over 18, how many adults over 18 did NOT report that they enjoyed the first book but not the second book?**

187 | |

352 | |

473 | |

626 |

Question 7 Explanation:

The correct answer is (C). Remember to find each category separately to keep the different numbers clear. The total number of people was 2,200.

$\frac{2}{5}$ of 2,200 = 880, so 880 of those surveyed said they did not enjoy the second book, but enjoyed the first book. Of these 880 people, 40% were adults over 18, so in this group there were 880 * 0.4 = 352 people.

It is also stated that $\frac{3}{8}$ of the 2,200 surveyed, or 825 people who are adults over 18. To find the number of adults who did not report that they enjoyed the first book but not the second, subtract the portion who did report they enjoyed the first book but not the second from the total number of adults:

825 − 352 = 473 adults.

$\frac{2}{5}$ of 2,200 = 880, so 880 of those surveyed said they did not enjoy the second book, but enjoyed the first book. Of these 880 people, 40% were adults over 18, so in this group there were 880 * 0.4 = 352 people.

It is also stated that $\frac{3}{8}$ of the 2,200 surveyed, or 825 people who are adults over 18. To find the number of adults who did not report that they enjoyed the first book but not the second, subtract the portion who did report they enjoyed the first book but not the second from the total number of adults:

825 − 352 = 473 adults.

Question 8 |

**If (**

*m*,ƒ(*m*)) represents a point on the graph ƒ(*m*) = 2*m*+ 1, which of the following could be a portion of the graph of the set of points (*m*,(ƒ(*m*))^{2})?Question 8 Explanation:

The correct answer is (C). Let’s re-write the “

Begin with the equation provided in the question, ƒ(

ƒ(

ƒ(

By factoring out a 4 from the expression, the vertex form of the quadratic can be found:

4(

The correct answer is (C). This function translates graphically into a parabola with a vertex at (−½, 0) that is vertically stretched and opens upwards. Only answer choice (C) shows an appropriate possibility.

*m*” as an “*x*” to better understand how this function would look on an*xy*-coordinate plane.Begin with the equation provided in the question, ƒ(

*m*), and square it, ƒ(*m*)^{2}:ƒ(

*m*) = 2x + 1ƒ(

*m*)^{2}= (2*x*+ 1)^{2}= 4*x*^{2}+ 4*x*+ 1By factoring out a 4 from the expression, the vertex form of the quadratic can be found:

4(

*x*^{2}+*x*+ $\frac{1}{4}$) = 4(*x*+ $\frac{1}{2}$)^{2})The correct answer is (C). This function translates graphically into a parabola with a vertex at (−½, 0) that is vertically stretched and opens upwards. Only answer choice (C) shows an appropriate possibility.

Question 9 |

**Parallelogram**

*QRST*has an area of 120 and its longest side (*QT*) is 24. The angle opposite the vertical is 30°, and the vertical is from*R*to point*U*, which lies along*QT*. What is the length of the hypotenuse of the triangle formed from segments*RU*,*QU*, and*QR*, rounded to the nearest whole number?5 | |

8 | |

10 | |

13 |

Question 9 Explanation:

The correct answer is (C). Since the area is 120 and the base is 24, we know from the question stem that the height (RU) is 5.

Given that the angle opposite the vertical is 30°, a 30°,60°,90° triangle should be seen. Recall that the ratio of the side lengths of a 30, 60, 90 triangle is:

$x:x\sqrt{3}:2x$

In this case, the smallest side length x is 5, so:

$5:5\sqrt{3}:2*5$

The hypotenuse is 2 * 5 = 10.

Given that the angle opposite the vertical is 30°, a 30°,60°,90° triangle should be seen. Recall that the ratio of the side lengths of a 30, 60, 90 triangle is:

$x:x\sqrt{3}:2x$

In this case, the smallest side length x is 5, so:

$5:5\sqrt{3}:2*5$

The hypotenuse is 2 * 5 = 10.

Question 10 |

**The initial number of elements in Set**

*A*is*x*, where*x*> 0. If the number of elements in Set*A*doubles every hour, which of the following represents the increase in the number of elements in Set*A*after exactly one day?23 x^{23} | |

(2 ^{23})x | |

23 x^{24} | |

(2 ^{24})x − x |

Question 10 Explanation:

The correct answer is (D). To find the INCREASE in the elements, we need to subtract the original number from the final total number. Since the original amount is going to be multiplied by 2 (or doubled) 24 times, we can express this as the exponent: 2

^{24}. The final total increase will be 2^{24}*x*(the final total) −*x*, (the original number).
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