Choice A is correct. An equation of a line of best fit for data set F can be written in the form \( y = a + bx \), where \( a \) is the \( y \)-coordinate of the \( y \)-intercept of the line of best fit and \( b \) is the slope. The line of best fit shown for data set E has a \( y \)-intercept at approximately \( (0, 12) \). It’s given that data set F is created by multiplying the \( y \)-coordinate of each data point from data set E by 3.9. It follows that a line of best fit for data set F has a \( y \)-intercept at approximately \( (0, 12(3.9)) \), or \( (0, 46.8) \). Therefore, the value of \( a \) is approximately 46.8. The slope of a line that passes through points \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \) can be calculated as \( \frac{y_{2} – y_{1}}{x_{2} – x_{1}} \). Since the line of best fit shown for data set E passes approximately through the point \( (12, 30) \), it follows that a line of best fit for data set F passes approximately through the point \( (12, 30(3.9)) \), or \( (12, 117) \). Substituting \( (0, 46.8) \) and \( (12, 117) \) for \( (x_{1}, y_{1}) \) and \( (x_{2}, y_{2}) \), respectively, in \( \frac{y_{2} – y_{1}}{x_{2} – x_{1}} \) yields \( \frac{117 – 46.8}{12 – 0} \), which is equivalent to \( \frac{70.2}{12} \), or 5.85. Therefore, the value of \( b \) is approximately 5.85, or approximately 5.9. Thus, \( y = 46.8 + 5.9x \) could be an equation of a line of best fit for data set F.

Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This could be an equation of a line of best fit for data set E, not data set F.

Choice A is correct. The equation \( f(x) = (1.84)^{\frac{x}{4}} \) can be rewritten as \( f(x) = (1.84)^{(\frac{1}{4})(x)} \), which is equivalent to \( f(x) = (1.84^{\frac{1}{4}})^{x} \), or approximately \( f(x) = (1.16467)^{x} \). Since it’s given that \( f(x) = (1.84)^{\frac{x}{4}} \) can be rewritten as \( f(x) = (1 + \frac{p}{100})^{x} \), where \( p \) is a constant, it follows that \( 1 + \frac{p}{100} \) is approximately equal to 1.16467. Therefore, \( \frac{p}{100} \) is approximately equal to 0.16467. It follows that the value of \( p \) is approximately equal to 16.467. Of the given choices, 16 is closest to the value of \( p \).

Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. It’s given that \( f(24) < 0 \). Substituting 24 for \( f(x) \) in the equation \( f(x) = a\sqrt{x + b} \) yields \( f(24) = a\sqrt{24 + b} \). Therefore, \( a\sqrt{24 + b} < 0 \). Since \( \sqrt{24 + b} \) can't be negative, it follows that \( a < 0 \). It's also given that the graph of \( y = f(x) \) passes through the point \( (-24, 0) \). It follows that when \( x = -24 \), \( f(x) = 0 \). Substituting \( -24 \) for \( x \) and 0 for \( f(x) \) in the equation \( f(x) = a\sqrt{x + b} \) yields \( 0 = a\sqrt{-24 + b} \). By the zero product property, either \( a = 0 \) or \( \sqrt{-24 + b} = 0 \). Since \( a < 0 \), it follows that \( \sqrt{-24 + b} = 0 \). Squaring both sides of this equation yields \( -24 + b = 0 \). Adding 24 to both sides of this equation yields \( b = 24 \). Since \( a < 0 \) and \( b \) is 24, it follows that \( a < b \) must be true. Choice A is incorrect. The value of \( f(0) \) is \( a\sqrt{b} \), which must be negative. Choice B is incorrect. The value of \( f(0) \) is \( a\sqrt{b} \), which could be \( -24 \), but doesn't have to be. Choice C is incorrect and may result from conceptual or calculation errors.

Choice B is correct. The Pythagorean theorem states that for a right triangle, \( c^{2} = a^{2} + b^{2} \), where \( c \) represents the length of the hypotenuse and \( a \) and \( b \) represent the lengths of the legs. It’s given that in triangle \( ABC \), angle \( B \) is a right angle. Therefore, triangle \( ABC \) is a right triangle, where the hypotenuse is side \( AC \) and the legs are sides \( AB \) and \( BC \). It’s given that the lengths of sides \( AB \) and \( BC \) are \( 10\sqrt{37} \) and \( 24\sqrt{37} \), respectively. Substituting these values for \( a \) and \( b \) in the formula \( c^{2} = a^{2} + b^{2} \) yields \( c^{2} = (10\sqrt{37})^{2} + (24\sqrt{37})^{2} \), which is equivalent to \( c^{2} = 100(37) + 576(37) \), or \( c^{2} = 676(37) \). Taking the square root of both sides of this equation yields \( c = \pm 26\sqrt{37} \). Since \( c \) represents the length of the hypotenuse, side \( AC \), \( c \) must be positive. Therefore, the length of side \( AC \) is \( 26\sqrt{37} \).

Choice A is incorrect. This is the result of solving the equation \( c = 24\sqrt{37} – 10\sqrt{37} \), not \( c^{2} = (10\sqrt{37})^{2} + (24\sqrt{37})^{2} \). Choice C is incorrect. This is the result of solving the equation \( c = 10\sqrt{37} + 24\sqrt{37} \), not \( c^{2} = (10\sqrt{37})^{2} + (24\sqrt{37})^{2} \). Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It’s given that there is a linear relationship between the number of cars, \( c \), on a commuter train and the maximum number of passengers and crew, \( p \), that the train can carry. It follows that this relationship can be represented by an equation of the form \( p = mc + b \), where \( m \) is the rate of change of \( p \) in this relationship and \( b \) is a constant. The rate of change of \( p \) in this relationship can be calculated by dividing the difference in any two values of \( p \) by the difference in the corresponding values of \( c \). Using two pairs of values given in the table, the rate of change of \( p \) in this relationship is \( \frac{284 – 174}{5 – 3} \), or 55. Substituting 55 for \( m \) in the equation \( p = mc + b \) yields \( p = 55c + b \). The value of \( b \) can be found by substituting any value of \( c \) and its corresponding value of \( p \) for \( c \) and \( p \), respectively, in this equation. Substituting 10 for \( c \) and 559 for \( p \) yields \( 559 = 55(10) + b \), or \( 559 = 550 + b \). Subtracting 550 from both sides of this equation yields \( 9 = b \). Substituting 9 for \( b \) in the equation \( p = 55c + b \) yields \( p = 55c + 9 \). Subtracting 9 from both sides of this equation yields \( p – 9 = 55c \). Subtracting \( p \) from both sides of this equation yields \( -9 = 55c – p \), or \( 55c – p = -9 \).

Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect and may result from conceptual or calculation errors.

Choice D is correct. Since the square of a real number is never negative, the given equation isn’t true for any real value of x. Therefore, the given equation has zero distinct real solutions.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice C is incorrect and may result from conceptual or calculation errors.

Choice A is correct. The volume, \( V \), of a right circular cylinder is given by the formula \( V = \pi r^{2}h \), where \( \pi r^{2} \) is the area of the base of the cylinder and \( h \) is the height. It’s given that a right circular cylinder has a volume of 432 cubic centimeters and the area of the base is 24 square centimeters. Substituting 432 for \( V \) and 24 for \( \pi r^{2} \) in the formula \( V = \pi r^{2}h \) yields \( 432 = 24h \). Dividing both sides of this equation by 24 yields \( 18 = h \). Therefore, the height of the cylinder, in centimeters, is 18.

Choice B is incorrect. This is the area of the base, in square centimeters, not the height, in centimeters, of the cylinder. Choice C is incorrect. This is the height, in centimeters, of a cylinder if its volume is 432 cubic centimeters and the area of its base is 2, not 24, cubic centimeters. Choice D is incorrect. This is the height, in centimeters, of a cylinder if its volume is 432 cubic centimeters and the area of its base is \( \frac{1}{24} \), not 24, cubic centimeters.

Choice B is correct. If a data set contains an even number of data values, when
the data values are listed in ascending or descending order, the median is
between the two middle values. The given data set contains 8 values. When listed
in ascending order, the data set is 4, 4, 4, 5, 5, 6, 10, 18 and the two middle values
are 5 and 5. Since the two middle values are the same, the median must be 5.
Choice A is incorrect. This value is between the two middle values in the list shown, not the two middle values when the data values are listed in ascending or descending order. Choice C is incorrect. This is the mean, not the median, of the data set. Choice D is incorrect. This is the range, not the median, of the data set.

The correct answer is 5. Let \( x \) represent the width, in inches, of the rectangle. It’s given that the length of the rectangle is 4 inches less than 7 times its width, or \( 7x – 4 \) inches. The area of a rectangle is equal to its width multiplied by its length. Multiplying the width, \( x \) inches, by the length, \( 7x – 4 \) inches, yields \( x(7x – 4) \) square inches. It’s given that the rectangle has an area of 155 square inches, so it follows that \( x(7x – 4) = 155 \), or \( 7x^{2} – 4x = 155 \). Subtracting 155 from both sides of this equation yields \( 7x^{2} – 4x – 155 = 0 \). Factoring the left-hand side of this equation yields \( (7x + 31)(x – 5) = 0 \). Applying the zero product property to this equation yields two solutions: \( x = -\frac{31}{7} \) and \( x = 5 \). Since \( x \) is the rectangle’s width, in inches, which must be positive, the value of \( x \) is 5. Therefore, the width of the rectangle, in inches, is 5.

Choice C is correct. Applying the zero product property to the given equation yields three equations: \( x + 2 = 0 \), \( x – 5 = 0 \), and \( x + 9 = 0 \). Subtracting 2 from both sides of the equation \( x + 2 = 0 \) yields \( x = -2 \). Adding 5 to both sides of the equation \( x – 5 = 0 \) yields \( x = 5 \). Subtracting 9 from both sides of the equation \( x + 9 = 0 \) yields \( x = -9 \). Therefore, the solutions to the given equation are \( -2 \), \( 5 \), and \( -9 \). It follows that a positive solution to the given equation is 5.

Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect and may result from conceptual or calculation errors.
Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. Dividing each side of the given equation by 7 yields \( \frac{7(2x – 3)}{7} = \frac{63}{7} \), or \( 2x – 3 = 9 \). Therefore, the equation \( 2x – 3 = 9 \) is equivalent to the given equation and has the same solution.

Choice B is incorrect. This equation is equivalent to \( 7(2x – 3) = 392 \), not \( 7(2x – 3) = 63 \). Choice C is incorrect. Distributing 7 on the left-hand side of the given equation yields \( 14x – 21 = 63 \), not \( 2x – 21 = 63 \). Choice D is incorrect. Distributing 7 on the left-hand side of the given equation yields \( 14x – 21 = 63 \), not \( 2x – 21 = 70 \).

Choice A is correct. It’s given that the ratio of the length of line segment \( XY \) to the length of line segment \( ZV \) is 6 to 1, which means \( \frac{XY}{ZV} = \frac{6}{1} \). It’s given that the length of line segment \( XY \) is 102 inches. If the length, in inches, of line segment \( ZV \) is represented by \( \ell \), the value of \( \ell \) can be calculated by solving the equation \( \frac{102}{\ell} = \frac{6}{1} \), or \( \frac{102}{\ell} = 6 \). Multiplying each side of this equation by \( \ell \) yields \( 102 = 6\ell \). Dividing each side of this equation by 6 yields \( 17 = \ell \). Therefore, the length of line segment \( ZV \) is 17 inches.

Choice B is incorrect. This is the length, in inches, of line segment \( ZV \) if the length of line segment \( XY \) is 576, not 102, inches. Choice C is incorrect. This is the length, in inches, of line segment \( XY \), not line segment \( ZV \). Choice D is incorrect. This is the length, in inches, of line segment \( ZV \) if the ratio of the length of line segment \( XY \) to the length of line segment \( ZV \) is 1 to 6, not 6 to 1.

Choice B is correct. Since \( x^{3} \) is a common factor of each term in the given expression, the expression can be rewritten as \( x^{3}(5x^{2} – 6x + 8) \).

Choice A is incorrect. This expression is equivalent to \( 5x^{5} – 6x^{4} \). Choice C is incorrect. This expression is equivalent to \( 40x^{5} – 48x^{4} + 8x^{3} \). Choice D is incorrect. This expression is equivalent to \( -36x^{9} + 48x^{8} + 6x^{5} \).

The correct answer is 10. It’s given that the amount of Hanna’s food order was $50 and that Hanna gave a tip of 20% of the amount of the bill. 20% of 50 can be calculated as \( (\frac{20}{100})(50) \), which yields \( \frac{1000}{100} \), or 10. Therefore, the amount, in dollars, of the tip Hanna gave is 10.

The correct answer is 6. The first equation in the given system is x=8.
Substituting 8 for x in the second equation in the given system yields
8 = 3y = 26. Subtracting 8 from both sides of this equation yields 3y= 18.
Dividing both sides of this equation by 3 yields y=6. Therefore, the value
of y is 6

Choice A is correct. The graph of function \( f \) shows that as \( x \) increases, \( f(x) \) also increases, which means \( f(x) \) is an increasing function. The graph of \( f \) is a line, which indicates a constant rate of change. A function that has a constant rate of change is a linear function. Therefore, function \( f \) can be described as increasing linear.

Choice B is incorrect. For a decreasing function, as \( x \) increases, \( f(x) \) decreases, rather than increases. Choice C is incorrect. The graph of an exponential function isn’t a line. Choice D is incorrect. For a decreasing function, as \( x \) increases, \( f(x) \) decreases, rather than increases, and the graph of an exponential function isn’t a line.

Choice B is correct. If a house from the street is selected at random, the probability of selecting a house that is blue is equal to the number of houses on the street that are blue divided by the total number of houses on the street. Since there are 2 blue houses on a street with 7 total houses, the probability of selecting a house that is blue from this street is \( \frac{2}{7} \).

Choice A is incorrect. This is the probability of selecting a house that is blue from a street on which 1 of the 7 houses is blue. Choice C is incorrect. This is the probability of selecting a house that is not blue from this street. Choice D is incorrect. This is the probability of selecting a house that is blue from a street on which all the houses are blue.

Choice A is correct. It’s given that angle 1 and angle 2 are vertical angles, and the measure of angle 1 is 72°. Vertical angles have equal measures. Therefore, the measure of angle 2 is 72°.
Choice B is incorrect. This is the measure of an angle that is supplementary, not congruent, to angle 1. Choice C is incorrect. This is the sum of the measures of angle 1 and angle 2. Choice D is incorrect and may result from conceptual or calculation errors

Choice B is correct. Substituting 72 for \( f(x) \) in the given function yields \( 72 = 8x \). Dividing each side of this equation by 8 yields \( 9 = x \). Therefore, \( f(x) = 72 \) when the value of \( x \) is 9.

Choice A is incorrect. This is the value of \( x \) for which \( f(x) = 64 \), not \( f(x) = 72 \). Choice C is incorrect. This is the value of \( x \) for which \( f(x) = 512 \), not \( f(x) = 72 \). Choice D is incorrect. This is the value of \( x \) for which \( f(x) = 640 \), not \( f(x) = 72 \).

Choice B is correct. The function P gives the estimated number of marine
mammals in a certain area, where t is the number of years since a study began.
Since the value of P^ h 0 is the value of P t^ h when t=0, it follows that P^ h 0 1 = ,800
means that the value of P t^ h is 1 8, 00 when t=0. Since t is the number of years
since the study began, it follows that t=0 is 0 years since the study began, or
when the study began. Therefore, the best interpretation of P^ h 0 1 = ,800 in this
context is the estimated number of marine mammals in the area was 1 8, 00 when
the study began.

The correct answer is 11. It’s given that the function \( f(x) = 14 + 4x \) represents the total cost, in dollars, of attending an arcade when \( x \) games are played. Substituting 58 for \( f(x) \) in the given equation yields \( 58 = 14 + 4x \). Subtracting 14 from each side of this equation yields \( 44 = 4x \). Dividing each side of this equation by 4 yields \( 11 = x \). Therefore, 11 games can be played for a total cost of $58.

**Choice D** is correct. Adding $53$ to each side of the given equation yields $k^2 = 144$. Taking the square root of each side of this equation yields $k = \pm 12$. Therefore, the positive solution to the given equation is $12$.

*Choice A* is incorrect. This is the positive solution to the equation $k^2 – 53 = 20,683$, not $k^2 – 53 = 91$. *Choice B* is incorrect. This is the positive solution to the equation $k^2 – 53 = 5,131$, not $k^2 – 53 = 91$. *Choice C* is incorrect. This is the positive solution to the equation $k^2 – 53 = 1,391$, not $k^2 – 53 = 91$.