**Choice C** is correct. It’s given that the circle has its center at \((-1, 1)\) and that line \(t\) is tangent to this circle at the point \((5, -4)\). Therefore, the points \((-1, 1)\) and \((5, -4)\) are the endpoints of the radius of the circle at the point of tangency. The slope of a line or line segment that contains the points \((a, b)\) and \((c, d)\) can be calculated as \(\frac{d – b}{c – a}\). Substituting \((-1, 1)\) for \((a, b)\) and \((5, -4)\) for \((c, d)\) in the expression \(\frac{d – b}{c – a}\) yields \(\frac{-4 – 1}{5 – (-1)}\), or \(-\frac{5}{6}\). Thus, the slope of this radius is \(-\frac{5}{6}\). A line that’s tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It follows that line \(t\) is perpendicular to the radius at the point \((5, -4)\), so the slope of line \(t\) is the negative reciprocal of the slope of this radius. The negative reciprocal of \(-\frac{5}{6}\) is \(\frac{6}{5}\). Therefore, the slope of line \(t\) is \(\frac{6}{5}\). Since the slope of line \(t\) is the same between any two points on line \(t\), a point lies on line \(t\) if the slope of the line segment connecting the point and \((5, -4)\) is \(\frac{6}{5}\). Substituting choice C, \((10, 2)\), for \((a, b)\) and \((5, -4)\) for \((c, d)\) in the expression \(\frac{d – b}{c – a}\) yields \(\frac{-4 – 2}{5 – 10}\), or \(\frac{6}{5}\). Therefore, the point \((10, 2)\) lies on line \(t\).

*Choice A* is incorrect. The slope of the line segment connecting \(\left(0, \frac{6}{5}\right)\) and \((5, -4)\) is \(\frac{-4 – \frac{6}{5}}{5 – 0}\), or \(-\frac{26}{25}\), not \(\frac{6}{5}\). *Choice B* is incorrect. The slope of the line segment connecting \((4, 7)\) and \((5, -4)\) is \(\frac{-4 – 7}{5 – 4}\), or \(-11\), not \(\frac{6}{5}\). *Choice D* is incorrect. The slope of the line segment connecting \((11, 1)\) and \((5, -4)\) is \(\frac{-4 – 1}{5 – 11}\), or \(\frac{5}{6}\), not \(\frac{6}{5}\).

**Choice A** is correct. It’s given that the function \(P\) models the population, in thousands, of a certain city \(t\) years after \(2003\). The value of the base of the given exponential function, \(1.04\), corresponds to an increase of \(4\%\) for every increase of \(1\) in the exponent, \(\left(\frac{6}{4}\right)t\). If the exponent is equal to \(0\), then \(\left(\frac{6}{4}\right)t = 0\). Multiplying both sides of this equation by \(\left(\frac{4}{6}\right)\) yields \(t = 0\). If the exponent is equal to \(1\), then \(\left(\frac{6}{4}\right)t = 1\). Multiplying both sides of this equation by \(\left(\frac{4}{6}\right)\) yields \(t = \frac{4}{6}\), or \(t = \frac{2}{3}\). Therefore, the population is predicted to increase by \(4\%\) every \(\frac{2}{3}\) of a year. It’s given that the population is predicted to increase by \(4\%\) every \(n\) months. Since there are \(12\) months in a year, \(\frac{2}{3}\) of a year is equivalent to \(\left(\frac{2}{3}\right)(12)\), or \(8\), months. Therefore, the value of \(n\) is \(8\).

*Choice B* is incorrect. This is the number of months in which the population is predicted to increase by \(4\%\) according to the model \(P(t) = 260(1.04)^t\), not \(P(t) = 260(1.04)^{\left(\frac{6}{4}\right)t}\). *Choice C* is incorrect. This is the number of months in which the population is predicted to increase by \(4\%\) according to the model \(P(t) = 260(1.04)^{\left(\frac{4}{6}\right)t}\), not \(P(t) = 260(1.04)^{\left(\frac{6}{4}\right)t}\). *Choice D* is incorrect. This is the number of months in which the population is predicted to increase by \(4\%\) according to the model \(P(t) = 260(1.04)^{\left(\frac{1}{6}\right)t}\), not \(P(t) = 260(1.04)^{\left(\frac{6}{4}\right)t}\).

Choice D is correct. The number of solutions to a quadratic equation in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, can be determined by the value of the discriminant, $b^2 – 4ac$. If the value of the discriminant is greater than zero, then the quadratic equation has two distinct real solutions. If the value of the discriminant is equal to zero, then the quadratic equation has exactly one real solution. If the value of the discriminant is less than zero, then the quadratic equation has no real solutions. For the quadratic equation in choice D, $5x^2 – 14x + 49 = 0$, $a = 5$, $b = -14$, and $c = 49$. Substituting 5 for $a$, -14 for $b$, and 49 for $c$ in $b^2 – 4ac$ yields $(-14)^2 – 4(5)(49)$, or -784. Since -784 is less than zero, it follows that the quadratic equation $5x^2 – 14x + 49 = 0$ has no real solutions.

Choice A is incorrect. The value of the discriminant for this quadratic equation is 392. Since 392 is greater than zero, it follows that this quadratic equation has two real solutions. Choice B is incorrect. The value of the discriminant for this quadratic equation is 0. Since zero is equal to zero, it follows that this quadratic equation has exactly one real solution. Choice C is incorrect. The value of the discriminant for this quadratic equation is 1,176. Since 1,176 is greater than zero, it follows that this quadratic equation has two real solutions.

*Choice A* is incorrect. This expression is equivalent to \(\frac{42a}{k} + \frac{42a}{k}\). *Choice B* is incorrect and may result from conceptual or calculation errors.

**Choice D** is correct. Two fractions can be added together when they have a common denominator. Since \(k > 0\), multiplying the second term in the given expression by \(\frac{k}{k}\) yields \(\frac{(42ak)k}{k}\), which is equivalent to \(\frac{42ak^2}{k}\). Therefore, the expression \(\frac{42a}{k} + 42ak\) can be written as \(\frac{42a}{k} + \frac{42ak^2}{k}\) which is equivalent to \(\frac{42a + 42ak^2}{k}\). Since each term in the numerator of this expression has a factor of \(42a\), the expression \(\frac{42a + 42ak^2}{k}\) can be rewritten as \(\frac{42a(1) + 42a(k^2)}{k}\), or \(\frac{42a(1 + k^2)}{k}\), which is equivalent to \(\frac{42a(k^2 + 1)}{k}\).

Choice D is correct. The area of a rectangle is given by $bh$, where $b$ is the length of the base of the rectangle and $h$ is its height. Let $x$ represent the length, in units, of the base of rectangle $ABCD$, and let $y$ represent its height, in units. Substituting $x$ for $b$ and $y$ for $h$ in the formula $bh$ yields $xy$. Therefore, the area, in square units, of $ABCD$ can be represented by the expression $xy$. It’s given that the length of each side of $EFGH$ is 6 times the length of the corresponding side of $ABCD$. Therefore, the length, in units, of the base of $EFGH$ can be represented by the expression $6x$, and its height, in units, can be represented by the expression $6y$. Substituting $6x$ for $b$ and $6y$ for $h$ in the formula $bh$ yields $(6x)(6y)$, which is equivalent to $36xy$. Therefore, the area, in square units, of $EFGH$ can be represented by the expression $36xy$. It’s given that the area of $ABCD$ is 54 square units. Since $xy$ represents the area, in square units, of $ABCD$, substituting 54 for $xy$ in the expression $36xy$ yields $36(54)$, or 1,944. Therefore, the area, in square units, of $EFGH$ is 1,944.

Choice A is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{\frac{1}{6}}$, not 6, times the length of the corresponding side of $ABCD$. Choice B is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{\frac{2}{3}}$, not 6, times the length of the corresponding side of $ABCD$. Choice C is incorrect. This is the area of a rectangle where the length of each side of the rectangle is $\sqrt{6}$, not 6, times the length of the corresponding side of $ABCD$.

Choice B is correct. Dividing each side of the given equation by 4 yields $\sqrt{2x} = 4$. Squaring both sides of this equation yields $2x = 16$. Multiplying each side of this equation by 3 yields $6x = 48$. Therefore, the value of $6x$ is 48.

Choice A is incorrect. This is the value of $3x$, not $6x$. Choice C is incorrect. This is the value of $9x$, not $6x$. Choice D is incorrect. This is the value of $12x$, not $6x$.

**Choice B** is correct. The given expression is equivalent to \(8x^3 + 8 – x^3 – (-2)\), or \(8x^3 + 8 – x^3 + 2\). Combining like terms in this expression yields \(7x^3 + 10\).

Choice A is correct. It’s given that the table shows an original data set of 5 values. It’s also given that a sixth value is added to create a new data set. The new data set consists of the 5 values in the original data set and one additional value, 121. Since the additional value, 121, is less than any value in the original data set, the mean of the original data set is greater than the mean of the new data set.

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 A is correct. The sum of the measures of the angles of a triangle is $180^\circ$. Therefore, the sum of the measures of $\angle R$, $\angle S$, and $\angle T$ is $180^\circ$. It’s given that the measure of $\angle R$ is $63^\circ$. It follows that the sum of the measures of $\angle S$ and $\angle T$ is $(180 – 63)^\circ$, or $117^\circ$. Therefore, the measure of $\angle S$, in degrees, must be less than 117. Of the given choices, only 116 is less than 117. Thus, the measure, in degrees, of $\angle S$ could be 116.

Choice B is incorrect. If the measure of $\angle S$ is $118^\circ$, then the sum of the measures of the angles of the triangle is greater than, not equal to, $180^\circ$. Choice C is incorrect. If the measure of $\angle S$ is $126^\circ$, then the sum of the measures of the angles of the triangle is greater than, not equal to, $180^\circ$. Choice D is incorrect. This is the sum of the measures of the angles of a triangle, in degrees.

Choice D is correct. It’s given that at a movie theater, there are a total of 350 customers and that each customer is located in either theater A, theater B, or theater C. If the probability of selecting a customer in theater A is 0.48, then $(0.48)(350)$, or 168, customers are located in theater A. If the probability of selecting a customer in theater B is 0.24, then $(0.24)(350)$, or 84, customers are located in theater B. It follows that there are $168 + 84$, or 252, customers in theater A and theater B. Therefore, there are $350 – 252$, or 98, customers in theater C.

Choice A is incorrect. This is the percent, not the number, of the customers that are located in theater C. Choice B is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the number of customers that are located in theater B, not theater C.

Choice C is correct. Any data point that’s located above the line of best fit has a y-value that’s greater than the y-value predicted by the line of best fit. For the scatterplot shown, 6 of the data points are above the line of best fit. Therefore, 6 of the data points have an actual y-value that’s greater than the y-value predicted by the line of best fit.
Choice A is incorrect and may result from conceptual or calculation errors.
Choice B is incorrect. This is the number of data points that have an actual y-value that’s less than the y-value predicted by the line of best fit. Choice D is incorrect and may result from conceptual or calculation errors.

Choice A is correct. It’s given that the kinetic energy, in joules, of an object with a mass of 9 kilograms traveling at a speed of $v$ meters per second is given by the function $K$, where $K(v) = \frac{9}{2}v^2$. It follows that in the equation $K(34) = 5,202$, 34 is the value of $v$, or the speed of the object, in meters per second, and 5,202 is the kinetic energy, in joules, of the object at that speed. Therefore, the best interpretation of $K(34) = 5,202$ in this context is the object traveling at 34 meters per second has a kinetic energy of 5,202 joules.

Choice B is incorrect. The object traveling at 340 meters per second has a kinetic energy of 520,200 joules. Choice C is incorrect. The object traveling at 5,202 meters per second has a kinetic energy of 121,773,618 joules. Choice D is incorrect. The object traveling at 23,409 meters per second has a kinetic energy of 2,465,915,764.5 joules.

Choice B is correct. It’s given that the graph shows the height above the water y,
in meters, of a diver x seconds after diving from a platform. The x-intercept of a
graph is the point at which the graph intersects the x-axis, or when the value of y
is 0. The graph shown intersects the x-axis between x=1 and x=2. In other
words, the diver is 0 meters above the water, or hits the water, between 1 and 2
seconds after diving from the platform. Of the given choices, only choice B
includes an interpretation where the diver hits the water between 1 and 2
seconds. Therefore, the best interpretation of the x-intercept of the graph is the
diver hits the water at 1 6. seconds.

Choice D is correct. A line in the $xy$-plane with a slope of $m$ and a $y$-intercept of $(0, b)$ can be defined by an equation in the form $y = mx + b$. It’s given that line $r$ has a slope of 4 and passes through the point $(0, 6)$. It follows that $m = 4$ and $b = 6$. Substituting 4 for $m$ and 6 for $b$ in the equation $y = mx + b$ yields $y = 4x + 6$. Therefore, the equation $y = 4x + 6$ defines line $r$.

Choice A is incorrect. This equation defines a line that has a slope of $-6$, not 4, and passes through the point $(0, 4)$, not $(0, 6)$. Choice B is incorrect. This equation defines a line that has a slope of 6, not 4, and passes through the point $(0, 4)$, not $(0, 6)$. Choice C is incorrect. This equation defines a line that passes through the point $(0, -6)$, not $(0, 6)$.

Choice D is correct. It’s given that a veterinarian recommends that each day the rabbit should eat 25 calories per pound of the rabbit’s weight, plus an additional 11 calories. If the rabbit’s weight is $x$ pounds, then multiplying 25 calories per pound by the rabbit’s weight, $x$ pounds, yields $25x$ calories. Adding the additional 11 calories that the rabbit should eat each day yields $25x + 11$ calories. It’s given that $c$ is the total number of calories the veterinarian recommends the rabbit should eat each day if the rabbit’s weight is $x$ pounds. Therefore, this situation can be represented by the equation $c = 25x + 11$.

Choice A is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat 25 calories per pound of the rabbit’s weight. Choice B is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat $25 + 11$, or 36, calories per pound of the rabbit’s weight. Choice C is incorrect. This equation represents a situation where a veterinarian recommends that each day the rabbit should eat 11 calories per pound of the rabbit’s weight, plus an additional 25 calories.

Choice B is correct. Since $2xy$ is a common factor of each term in the given expression, the expression can be rewritten as $2xy(8x^2y + 7)$.

Choice A is incorrect. This expression is equivalent to $16x^2y^2 + 14xy$. Choice C is incorrect. This expression is equivalent to $28x^3y^2 + 14xy$. Choice D is incorrect. This expression is equivalent to $112x^3y^2 + 14xy$.

Choice B is correct. It’s given that the function $f$ is defined by $f(x) = 4x – 3$. Substituting 10 for $x$ in the given function yields $f(10) = 4(10) – 3$, which is equivalent to $f(10) = 40 – 3$, or $f(10) = 37$. Therefore, the value of $f(10)$ is 37.

Choice A is incorrect and may result from conceptual or calculation errors. Choice C is incorrect. This is the value of $f(10)$ for the function $f(x) = 4x$, not $f(x) = 4x – 3$. Choice D is incorrect. This is the value of $f(10)$ for the function $f(x) = 4x + 3$, not $f(x) = 4x – 3$.

Choice B is correct. For the graph shown, the x-axis represents temperature, in kelvins, and the y-axis represents the estimated pressure, in pounds per square inch (psi) . The estimated pressure of the argon when the temperature is 600 kelvins can be found by locating the point on the graph where the value of x is equal to 600. The graph passes through the point (600 , 12) . This means that when the temperature is 600 kelvins, the estimated pressure is 12 psi.
Choice A is incorrect. This is the estimated pressure, in psi, of the argon when the temperature is 300 kelvins, not 600 kelvins. Choice C is incorrect and may result from conceptual or calculation errors. Choice D is incorrect. This is the temperature, in kelvins, of the argon.

Choice B is correct. 20% of 440 can be calculated as $(\frac{20}{100})(440)$, which is equivalent to $\frac{8,800}{100}$, or 88.

Choice A is incorrect. This is 10%, not 20%, of 440. Choice C is incorrect. This is 200%, not 20%, of 440. Choice D is incorrect. This is 400%, not 20%, of 440.

Choice D is correct. The volume of a right rectangular prism can be represented
by a function V that gives the volume of the prism, in cubic inches, in terms of the
length of the prism’s base. The volume of a right rectangular prism is equal to the
area of its base times its height. It’s given that the length of the prism’s base is
x inches, which is 7 inches more than the width of the prism’s base. This means
that the width of the prism’s base is x-7 inches. It follows that the area of the
prism’s base, in square inches, is x x^ h -7 and the volume, in cubic inches, of the
prism is x x^ ^ -7 9 h h. Thus, the function V that gives the volume of this right
rectangular prism, in cubic inches, in terms of the length of the prism’s base, x, is
V x^ ^ h h = – 9 7 x x .
Choice A is incorrect. This function would give the volume of the prism if the
height were 9 inches more than the length of its base and the width of the base
were 7 inches more than its length. Choice B is incorrect. This function would give
the volume of the prism if the height were 9 inches more than the length of its
base. Choice C is incorrect. This function would give the volume of the prism if the
width of the base were 7 inches more than its length, rather than the length of the
base being 7 inches more than its width.

Choice \( D \) is correct. A function of the form \( f(x) = a(b)^x + c \), where \( a < 0 \) and \( b > 1 \), is a decreasing function. Both of the given functions are of this form; therefore, both are decreasing functions. If a function \( f \) is decreasing, as the value of \( x \) increases, the corresponding value of \( f(x) \) decreases; therefore, the function doesn’t have a minimum value. Thus, neither of the given functions has a minimum value.

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. It is given that the result of increasing the quantity \( x \) by \( 400\% \) is \( 60 \). This can be written as \( x + \left(\frac{400}{100}\right)x = 60 \), which is equivalent to \( x + 4x = 60 \), or \( 5x = 60 \). Dividing each side of this equation by \( 5 \) yields \( x = 12 \). Therefore, the value of \( x \) is \( 12 \).

Choice \( B \) is incorrect. The result of increasing the quantity \( 15 \) by \( 400\% \) is \( 75 \), not \( 60 \).

Choice \( C \) is incorrect. The result of increasing the quantity \( 240 \) by \( 400\% \) is \( 1,200 \), not \( 60 \).

Choice \( D \) is incorrect. The result of increasing the quantity \( 340 \) by \( 400\% \) is \( 1,700 \), not \( 60 \).