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## Quiz #106

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Question 1 |

**A birthday cake with a height of 4 inches is cut into two pieces such that each piece is of a different size. If the ratio of the volume of the larger slice to the volume of the smaller slice is 5 to 3, what is the degree measure of the cut made into the cake?**

115° | |

120° | |

135° | |

145° |

Question 1 Explanation:

The correct answer is (C). Remember that the ratio between the volumes of the two pieces will be the same as the ratio of the areas of their bases, and also that the ratio between the interior angle of a sector of a circle and 360 degrees is the same as the ratio between the area of a sector and the area of an entire circle.

Since the ratio of the larger slice to the smaller slice is 5 to 3, the ratio of the area of the smaller slice to the area of the entire cake must be 3 to 8. This ratio is the same as the ratio of the interior angle of the sector representing the smaller slice to 360 degrees. We can therefore set up a proportion:

$\frac{3}{8} = \frac{x}{360}$

360(3) = 8

1080 = 8

135 =

Since the ratio of the larger slice to the smaller slice is 5 to 3, the ratio of the area of the smaller slice to the area of the entire cake must be 3 to 8. This ratio is the same as the ratio of the interior angle of the sector representing the smaller slice to 360 degrees. We can therefore set up a proportion:

$\frac{3}{8} = \frac{x}{360}$

360(3) = 8

*x*1080 = 8

*x*135 =

*x*Question 2 |

**Line**

*x*can be described by the function ƒ(x) = 5*x*. Line*y*is parallel to Line x such that the shortest distance between Line*y*and Line*x*is 5, and the*y*-intercept of Line*y*is negative. What is a possible equation for line*y*?$f(x) = x - 5$ | |

$f(x) + 5\sqrt{2} = 5x$ | |

$f(x) = x - 5\sqrt{2}$ | |

$f(x) - 5 = 5x$ |

Question 2 Explanation:

The correct answer is (B). Start by drawing the lines:

The slope-intercept form of a line is

The distance between Line

The slope-intercept form of a line is

*y*=*mx*+*b*, where*m*is the slope and*b*is the*y*-intercept. Parallel lines have the same slope, so Line*y*must also have a slope of 5; therefore, we can eliminate choices (A) and (C).The distance between Line

*x*and Line*y*is 5. If we drew a perpendicular line from the origin to Line*y*, we can form a right triangle with the*y*-axis as the hypotenuse and the distance between the lines as one of the legs. Since the hypotenuse is longer than either of the sides in a triangle, the*y*-intercept of Line*y*must be greater than 5. This eliminates choice (D).Question 3 |

$\dfrac{41}{12}$ | |

$\dfrac{41}{11}$ | |

$\dfrac{31}{9}$ | |

$\dfrac{31}{7}$ |

Question 3 Explanation:

The correct answer is (B). Use the given information to relate the values on the left with the values on the right. Remember that if two lines are divided proportionally, the corresponding segments are in proportion and the two lines and either pair of corresponding segments are in proportion. Set up the proportion and solve for the unknown:

$\dfrac{2y + 3}{5y - 4} = \dfrac{5}{7}$

Cross multiply and combine like terms to solve for

7(2

14

41 = 11

$y = \frac{41}{11}$

$\dfrac{2y + 3}{5y - 4} = \dfrac{5}{7}$

Cross multiply and combine like terms to solve for

*y*:7(2

*y*+ 3) = 5(5*y*− 4)14

*y*+ 21 = 25*y*− 2041 = 11

*y*$y = \frac{41}{11}$

Question 4 |

**VitaDrink contains 30 percent concentrated nutrients by volume. EnergyPlus contains 40 percent concentrated nutrients by volume. Which of the following represents the percent of concentrated nutrients by volume in a mixture of**

*v*gallons of VitaDrink,*e*gallons of EnergyPlus, and*w*gallons of water?$\dfrac{v+e}{v+e+w}$ | |

$\dfrac{0.3v+0.4e}{v+e+w}$ | |

$\dfrac{3v+4e}{v+e+w}$ | |

$\dfrac{30v+40e}{v+e+w}$ |

Question 4 Explanation:

The correct answer is (D). The total number of gallons in the final mixture will be the sum of all the components:

$100 * \dfrac{0.3v + 0.4e}{v + e + w} = \dfrac{30v + 40e}{v + e + w}$

If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture!

*v*+*e*+*w*. There are 0.3*v*gallons of nutrients from VitaDrink in the mixture, 0.4*e*gallons of nutrients from EnergyPlus in the mixture, and no nutrients from the water. The total number of gallons of nutrients in the new mixture will be 0.3*v*+ 0.4*e*. To convert from a fraction to a percent, we simply multiply our value by 100:$100 * \dfrac{0.3v + 0.4e}{v + e + w} = \dfrac{30v + 40e}{v + e + w}$

If you chose (B), remember that the question was asking for the percent, not the actual number in the mixture!

Question 5 |

**On a coordinate plane, (**

*a*,*b*) and (*a*+ 5,*b*+*c*), and (13, 10) are three points on line*l*. If the*x*-intercept of line*l*is −7, what is the value of*c*?1.5 | |

2.0 | |

2.5 | |

3.0 |

Question 5 Explanation:

The correct answer is (C). Recall that all lines can be written in the form

Using this revised equation in conjunction with the given points, we can first solve for the slope of the line in terms of

$slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(b+c)-b}{(a+5)-a} = \frac{c}{5}$

Substitute this value for the slope into the linear equation:

$b = \frac{c}{5} * a + k$

Notice that the question only asks for the value of c, which is a part of the slope of the line. If we use the given information to determine the actual value of the slope, we can equate the expression containing c with the actual value and solve for c. Given that the line has an x-intercept of −7, we can deduce that (−7, 0) is a point on the line. Calculate the slope of the line using this point and the point (13,10):

$slope = \frac{10 - 0}{13-(-7)} = \frac{10}{20} = \frac{1}{2}$

Equate this value with the expression containing

$\frac{1}{2} = \frac{c}{5}$

$c = 5 * \frac{1}{2} = 2.5$

*y*=*mx*+*b*, where m is the slope of the line (rise/run), and*b*is the*y*-intercept of the line. Given that the coordinate plane uses the variables*a*and*b*for*x*and*y*, we can rewrite the line equation as:*b*=*ma*+*k*, where the variable*k*is used to replace the original variable*b*for the*y*-intercept to avoid confusion.Using this revised equation in conjunction with the given points, we can first solve for the slope of the line in terms of

*c*. Given that the slope of a line is the change in the*y*variable divided by the change in the*x*variable, calculate the line’s slope:$slope = \frac{y_2 - y_1}{x_2 - x_1} = \frac{(b+c)-b}{(a+5)-a} = \frac{c}{5}$

Substitute this value for the slope into the linear equation:

$b = \frac{c}{5} * a + k$

Notice that the question only asks for the value of c, which is a part of the slope of the line. If we use the given information to determine the actual value of the slope, we can equate the expression containing c with the actual value and solve for c. Given that the line has an x-intercept of −7, we can deduce that (−7, 0) is a point on the line. Calculate the slope of the line using this point and the point (13,10):

$slope = \frac{10 - 0}{13-(-7)} = \frac{10}{20} = \frac{1}{2}$

Equate this value with the expression containing

*c*:$\frac{1}{2} = \frac{c}{5}$

$c = 5 * \frac{1}{2} = 2.5$

Question 6 |

**A bag contains 80% yellow marbles and 20% turquoise marbles. What is the probability, approximately, of obtaining exactly two turquoise marbles out of three randomly selected marbles?**

3.2% | |

8% | |

9.6% | |

18% |

Question 6 Explanation:

The correct answer is (C). The probability of getting one yellow marble is $\frac{4}{5}$.

The probability of getting one turquoise marble is $\frac{1}{5}$.

The probability of picking two turquoise marbles is $(\frac{1}{5})(\frac{1}{5})$ and picking one yellow marble is $(\frac{4}{5})$, and there are 3 ways of doing this:

Turquoise, Turquoise, Yellow

Turquoise, Yellow, Turquoise

Yellow, Turquoise, Turquoise

$(\frac{1}{5}) (\frac{1}{5}) (\frac{4}{5}) (3) = \frac{(1)(1)(4)(3)}{(5)(5)(5)}$

$\frac{12}{125}$ = 9.6%

The probability of getting one turquoise marble is $\frac{1}{5}$.

The probability of picking two turquoise marbles is $(\frac{1}{5})(\frac{1}{5})$ and picking one yellow marble is $(\frac{4}{5})$, and there are 3 ways of doing this:

Turquoise, Turquoise, Yellow

Turquoise, Yellow, Turquoise

Yellow, Turquoise, Turquoise

$(\frac{1}{5}) (\frac{1}{5}) (\frac{4}{5}) (3) = \frac{(1)(1)(4)(3)}{(5)(5)(5)}$

$\frac{12}{125}$ = 9.6%

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