1. \(k + 12 = 336\). What is the solution to the given equation?
Explanation: Subtract 12 from both sides: \(k = 336 - 12 = 348\).
2. The function \(f\) is defined by \(f(x) = x^3 + 15\). What is the value of \(f(2)\)?
Explanation: Substitute \(x = 2\): \(f(2) = 2^3 + 15 = 8 + 15 = 23\).
3. Sean rents a tent at a cost of $11 per day plus a onetime insurance fee of $10. Which equation represents the total cost \(c\), in dollars, to rent the tent with insurance for 4 days?
Explanation: The total cost includes \(4 \times 11\) for the rental and a $10 insurance fee.
4. In the figure shown, line \(m\) is parallel to line \(n\). What is the value of \(x\)?
Explanation: Use the parallel line angle relationships to determine that \(x = 26\).
5. John paid a total of $165 for a microscope by making a down payment of $37 plus \(p\) monthly payments of $16 each. Which of the following equations represents this situation?
Explanation: Total cost is \(37 + 16p = 165\), so solve for \(p\).
6. If \(y = 5x + 10\), what is the value of \(y\) when \(x = 8\)?
Explanation: Substitute \(x = 8\) into \(y = 5x + 10\): \(y = 5(8) + 10 = 40 + 10 = 50\).
7. The bar graph shows the distribution of 419 cans collected by 10 different groups for a food drive. How many cans were collected by group 6?
Explanation: According to the bar graph, group 6 collected 50 cans.
8. The table gives the distribution of votes for a new school mascot and grade level for 80 students. If one of these students is selected at random, what is the probability of selecting a student whose vote for the new mascot was for a lion?
Explanation: The total votes for the lion are 20. The total number of votes is 80. The probability is \(\frac{20}{80} = \frac{1}{4}\).
9. The graph represents the total charge, in dollars, by an electrician for \(x\) hours of work. The electrician charges a onetime fee plus an hourly rate. What is the best interpretation of the slope of the graph?
Explanation: The slope of a graph in a context like this represents the rate of change, which in this case is the hourly rate charged by the electrician.
10. Square \(X\) has a side length of 12 centimeters. The perimeter of square \(Y\) is 2 times the perimeter of square \(X\). What is the length, in centimeters, of one side of square \(Y\)?
Explanation: The perimeter of square \(X\) is \(4 \times 12 = 48\). The perimeter of square \(Y\) is \(2 \times 48 = 96\). Since the perimeter of a square is \(4 \times\) the side length, the side length of square \(Y\) is \(96 \div 4 = 14\).
11. What is the equation of the line that passes through the point (0, 5) and is parallel to the graph of \(y = 7x + 4\) in the \(xy\)-plane?
Explanation: A line parallel to \(y = 7x + 4\) has the same slope, which is 7. Using the point (0, 5), the equation is \(y = 7x + 5\).
12. In the linear function \(h\), \(h(0) = 41\) and \(h(1) = 40\). Which equation defines \(h\)?
Explanation: The slope of the line is \(-1\) since \(h(1) - h(0) = 40 - 41 = -1\). The equation is \(h(x) = -x + 41\), but since \(h(0) = 41\), it simplifies to \(h(x) = -x\).
13. The function \(f(t) = 60,000(2)^{\frac{t}{410}}\) gives the number of bacteria in a population \(t\) minutes after an initial observation. How much time, in minutes, does it take for the number of bacteria in the population to double?
Explanation: The doubling time is determined by the base \(2\) in the exponential function. It takes \(t = 410\) minutes for the population to double.
14. The function \(f(x) = (x - 6)(x - 2)(x + 6)\). In the \(xy\)-plane, the graph of \(y = g(x)\) is the result of translating the graph of \(y = f(x)\) up 4 units. What is the value of \(g(0)\)?
Explanation: Substituting \(x = 0\) into \(f(x)\), we find \(f(0) = (0 - 6)(0 - 2)(0 + 6) = -6 \cdot -2 \cdot 6 = -72\). Translating \(f(x)\) up 4 units gives \(g(0) = f(0) + 4 = -72 + 4 = -68\).
15. A candle is made of 17 ounces of wax. When the candle is burning, the amount of wax in the candle decreases by 1 ounce every 4 hours. If 16 ounces of wax remain in this candle, for how many hours has it been burning?
Explanation: The candle has burned \(17 - 16 = 1\) ounce of wax. Since the candle burns 1 ounce every 4 hours, it has been burning for \(4 \times 1 = 4\) hours.
16. The given equation \(14j + 5k = m\). Which equation correctly expresses \(k\) in terms of \(j\) and \(m\)?
Explanation: Solving for \(k\) in \(14j + 5k = m\), subtract \(14j\) from both sides to get \(5k = m - 14j\). Dividing through by 5 yields \(k = \frac{m - 14j}{5}\).
17. Triangle \(FGH\) is similar to triangle \(JKL\), where angle \(F\) corresponds to angle \(J\) and angles \(G\) and \(K\) are right angles. If \(\sin(F) = \frac{308}{317}\), what is the value of \(\sin(J)\)?
Explanation: Since triangles \(FGH\) and \(JKL\) are similar, their corresponding angles are congruent. Therefore, \(\sin(F) = \sin(J)\), and \(\sin(J) = \frac{308}{317}\).
18. The product of two positive integers is 546. If the first integer is 11 greater than twice the second integer, what is the smaller of the two integers?
Explanation: Let the smaller integer be \(x\). Then the larger integer is \(2x + 11\). The equation becomes \(x(2x + 11) = 546\), which simplifies to \(2x^2 + 11x - 546 = 0\). Solving this quadratic equation gives \(x = 39\) or a negative value, which is not valid.
19. \(y \leq x + 7\) and \(y \geq -2x - 1\). Which point \((x, y)\) is a solution to the given system of inequalities in the xy-plane?
Explanation: Substituting \((0, 14)\) into \(y \leq x + 7\) gives \(14 \leq 7\), which is true. Substituting \((0, 14)\) into \(y \geq -2x - 1\) gives \(14 \geq -1\), which is also true. The point satisfies both inequalities.
20. \(\sqrt{(x - 2)^2} = \sqrt{3x + 34}\). What is the smallest solution to the given equation?
Explanation: Square both sides to get \((x - 2)^2 = 3x + 34\). Expanding and rearranging gives \(x^2 - 7x - 30 = 0\). Factoring gives \((x - 10)(x + 3) = 0\), so \(x = 10\) or \(x = -3\). Only \(x = 2\) satisfies the original equation.
21. The regular price of a shirt at a store is $11.70. The sale price of the shirt is 80% less than the regular price, and the sale price is 30% greater than the store’s cost for the shirt. What was the store’s cost, in dollars, for the shirt?
Explanation: The sale price is \(11.70 \times 0.2 = 2.34\). Let the store's cost be \(c\). The sale price is \(1.3c\), so \(1.3c = 2.34\). Solving for \(c\) gives \(c = \frac{2.34}{1.3} \approx 2.77\).
22. A sample of oak has a density of 807 kilograms per cubic meter. The sample is in the shape of a cube, where each edge has a length of 0.90 meters. To the nearest whole number, what is the mass, in kilograms, of this sample?
Explanation: The volume of the cube is \(0.9^3 = 0.729\) cubic meters. The mass is given by density times volume: \(807 \times 0.729 = 897\) kilograms (rounded).
23. For \(x > 0\), the function \(f\) is defined as follows: \(f(x)\) equals 201% of \(x\). Which of the following could describe this function?
Explanation: The function \(f(x) = 2.01x\) is a linear function because it is proportional to \(x\) with a constant rate of increase (slope = 2.01).
24. The rational function \(f\) is defined by an equation in the form \(f(x) = \frac{a}{x + b}\), where \(a\) and \(b\) are constants. The partial graph of \(y = f(x)\) is shown. If \(g(x) = f(x + 4)\), which equation could define function \(g\)?
Explanation: The transformation \(f(x + 4)\) shifts the function horizontally, which modifies the denominator in the function's equation. The equation must account for the shift while retaining the proportionality factor \(a\).
25. Which expression is equivalent to \(\frac{y + 12}{x - 8} \cdot \frac{y(x - 8)}{x^2 y - 8xy + x - 8}\)?
Explanation: Simplify the given expression by canceling out common terms and carefully evaluating the numerators and denominators. The resulting expression matches option D.
26. The table shows the results of a poll. A total of 803 voters selected at random were asked which candidate they would vote for in the upcoming election. According to the poll, if 6,424 people vote in the election, by how many votes would Angel Cruz be expected to win?
Poll Results
Angel Cruz: 483
Terry Smith: 320
Explanation: To calculate Angel Cruz's expected win, find the proportion of votes each candidate received in the sample and scale it to 6,424 voters. Cruz's proportion: \( \frac{483}{803} \). Smith's proportion: \( \frac{320}{803} \). Multiply these by 6,424 and subtract the two totals to get Cruz's expected win.
27. The graph of \(x^2 + x + y^2 + y = \frac{199}{2}\) in the xy-plane is a circle. What is the length of the circle’s radius?
Explanation: Rewrite the equation \(x^2 + x + y^2 + y = \frac{199}{2}\) in standard form for a circle by completing the square for \(x\) and \(y\). This results in a circle with a radius \(r = \sqrt{199/2} \approx 15\).
1. Isabel grows potatoes in her garden. This year, she harvested 760 potatoes and saved 10% of them to plant next year. How many of the harvested potatoes did Isabel save to plant next year?
Explanation: 10% of 760 is 76, and Isabel saved this amount for planting next year.
2. What is the y-intercept of the graph shown?
Explanation: The graph crosses the y-axis at (0, 2).
3. What length, in centimeters, is equivalent to a length of 51 meters? (1 meter = 100 centimeters)
Explanation: 51 meters × 100 centimeters/meter = 51,000 centimeters.
4. A bus is traveling at a constant speed along a straight portion of road. The equation d = 30t gives the distance d, in feet from a road marker, that the bus will be t seconds after passing the marker. How many feet from the marker will the bus be 2 seconds after passing the marker?
Explanation: Using the equation d = 30 × 2 = 60 feet.
5. A car travels 300 miles using 10 gallons of gas. How many miles per gallon does the car get?
Explanation: Miles per gallon = 300 miles ÷ 10 gallons = 30 miles per gallon.
6. A square has an area of 64 square inches. What is the length of each side of the square?
Explanation: The side of the square is √64 = 8 inches.
7. What is the greatest common factor (GCF) of 36 and 48?
Explanation: The GCF of 36 and 48 is 12.
8. A recipe calls for 3 cups of flour and 2 cups of sugar. If you want to double the recipe, how many cups of sugar will you need?
Explanation: Doubling the recipe requires 2 × 3 cups = 6 cups of sugar.
9. If a train is traveling at a constant speed of 45 miles per hour, how far will it travel in 2 hours?
Explanation: Distance = Speed × Time, so 45 miles/hour × 2 hours = 90 miles.
10. If 5 pencils cost $2.50, what is the cost of 1 pencil?
Explanation: Cost of 1 pencil = $2.50 ÷ 5 pencils = $0.50 per pencil.
11. The scatterplot shows the relationship between two variables, x and y. A line of best fit is also shown.
Which of the following equations best represents the line of best fit shown?
Explanation: The slope and y-intercept of the line match this equation.
12. The function f is defined by \( f(x) = \frac{8}{\sqrt{x}} \). For what value of x does \( f(x) = 48 \)?
Explanation: Solve \( \frac{8}{\sqrt{x}} = 48 \) by isolating \( \sqrt{x} \) and squaring both sides.
13. A circle has center O, and points R and S lie on the circle. In triangle ORS, the measure of \( \angle ROS \) is 88°. What is the measure of \( \angle RSO \), in degrees?
Explanation: Use the properties of a triangle (sum of angles equals 180°).
14. Solve \( x(x + 1) - 56 = 4x(x - 7) \). What is the sum of the solutions to the given equation?
Explanation: Solve by expanding and combining terms, then find the roots and their sum.
15. The solution to the given system of equations is \( (x, y) \). What is the value of \( 5x \)?
\( y = 3x \)
\( 2x + y = 12 \)
Explanation: Substitute \( y = 3x \) into \( 2x + y = 12 \) and solve for x. Then, calculate \( 5x \).
16. A square has a side length of \( 4x \). What is the area of the square?
Explanation: Area of a square = \( \text{side}^2 \). So, \( (4x)^2 = 16x^2 \).
17. A bag contains 5 red, 4 blue, and 3 green marbles. If one marble is randomly selected, what is the probability that it is blue?
Explanation: Total marbles = 5 + 4 + 3 = 12. Probability of blue = \( \frac{4}{12} = \frac{1}{4} \).
18. A function is defined as \( g(x) = 3x - 7 \). If \( g(a) = 11 \), what is the value of \( a \)?
Explanation: Solve \( 3a - 7 = 11 \) for \( a \). Add 7 to both sides, then divide by 3.
19. A rectangle has a length that is twice its width. If the perimeter is 24 units, what is the width?
Explanation: Let the width = \( w \). Then, length = \( 2w \). Solve \( 2(w + 2w) = 24 \) for \( w \).
20. If \( x + y = 8 \) and \( x - y = 2 \), what is the value of \( x \)?
Explanation: Add the two equations to eliminate \( y \), then solve for \( x \).
21. The graph of \(9x - 10y = 19\) is translated down 4 units in the xy-plane. What is the x-coordinate of the x-intercept of the resulting graph?
Explanation: Translate the graph down by subtracting 4 from the y-intercept, then calculate the new x-intercept.
22. Two variables, \(x\) and \(y\), are related such that for each increase of 1 in the value of \(x\), the value of \(y\) increases by a factor of 4. When \(x = 0\), \(y = 200\). Which equation represents this relationship?
Explanation: The value of \(y\) is multiplied by 4 for each increase of 1 in \(x\), and the initial value of \(y\) is 200 when \(x = 0\).
23. One solution to the given equation can be written as \(1 + \sqrt{k}\), where \(k\) is a constant. What is the value of \(k\)?
Explanation: Solve the quadratic equation \(x^2 - 2x - 9 = 0\) and express the roots in the form \(1 + \sqrt{k}\).
24. The dot plots represent the distributions of values in data sets A and B. Which of the following statements must be true?
- I. The median of data set A is equal to the median of data set B.
- II. The standard deviation of data set A is equal to the standard deviation of data set B.
Explanation: Analyze the median and calculate or compare the variability of each distribution.
25. An isosceles right triangle has a perimeter of \(94 + 94\sqrt{2}\) inches. What is the length, in inches, of one leg of this triangle?
Explanation: Use the perimeter formula for an isosceles right triangle and solve for the leg's length.
26. In the given equation, \(c\) is a constant. The equation has exactly one solution. What is the value of \(c\)?
\(-9x^2 + 30x + c = 0\)
Explanation: For the quadratic equation to have exactly one solution, the discriminant must equal 0.
27. In the given system of equations, \(p\) is a constant. If the system has no solution, what is the value of \(p\)?
\[ \frac{3}{2}y - \frac{1}{4}x = \frac{2}{3} \quad \text{and} \quad \frac{1}{2}x + \frac{3}{2}y = p + \frac{9}{2} \]
Explanation: The system has no solution when the two equations represent parallel lines. This occurs when their slopes are equal, but their intercepts are different. Solve for \(p\) accordingly.
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