If x is the average (arithmetic mean) of m and 9, y is the average of 2m and 15, and z is the average of 3m and 18, what is the average of x, y, and z in terms of m ?

Solution
Since the average of 2 numbers is the sum of the 2 numbers divided by 2, the equations x = \(\frac{m+9}{2}\), y = \(\frac{2m + 15}{2}\) and z = \(\frac{3m + 18}{2}\) 2 are true. The average of x, y, and z is given by \(\frac{x + y + z}{3}\). Substituting the preceding expressions in m for each variable gives \(\frac{\frac{m + 9}{2}+\frac{2m + 15}{2}+\frac{3m + 18}{2}}{3}\). This fraction can be simplified to \(\frac{6m + 42}{6}\), or m + 7.
Choices A, C, and D are incorrect and may be the result of conceptual errors or calculation errors. For example, choice D is the sum of x, y, and z, not the average.
f(x) = (x + 6)(x − 4)
Which of the following is an equivalent form of the function f above in which the minimum value of f appears as a constant or coefficient?

Solution
The minimum value of a quadratic function appears as a constant in the vertex form of its equation, which can be found from the standard form by completing the square. Rewriting f(x) = (x + 6)(x − 4) in standard form gives f(x) = x^{2} + 2x − 24. Since the coefficient of the linear term is 2, the equation for f(x) can be rewritten in terms of (x + 1)^{2} as follows:
f(x) = x^{2} + 2x − 24 = (x^{2} + 2x + 1) − 1 − 24 = (x + 1)^{2} − 25
Since the square of a real number is always nonnegative, the vertex form f(x) = (x + 1)^{2} − 25 shows that the minimum value of f is −25 (and occurs at x = −1). Therefore, this equivalent form of f shows the minimum value of f as a constant.
Choices A and C are incorrect because they are not equivalent to the given equation for f. Choice B is incorrect because the minimum value of f, which is −25, does not appear as a constant or a coefficient.
The relative housing cost for a US city is defined to be the ratio \(\frac{average\, housing\, cost\, for\, the\, city}{national\, average\, housing\, cost}\), expressed as a percent.
The scatterplot above shows the relative housing cost and the population density for several large US cities in the year 2005. The line of best fit is also shown and has equation y x = 0.0125 + 61. Which of the following best explains how the number 61 in the equation relates to the scatterplot?

Solution
To interpret what the number 61 in the equation of the line of best fit represents, one must first understand what the data in the scatterplot represent. Each of the points in the scatterplot represents a large US city, graphed according to its population density (along the horizontal axis) and its relative housing cost (along the vertical axis). The line of best fit for this data represents the expected relative housing cost for a certain population density, based on the data points in the graph. Thus, one might say, on average, a city of population density x is expected to have a relative housing cost of y%, where y = 0.0125x + 61. The number 61 in the equation represents the yintercept of the line of best fit, in that when the population density, x, is 0, there is an expected relative housing cost of 61%. This might not make the best sense within the context of the problem, in that when the population density is 0, the population is 0, so there probably wouldn’t be any housing costs. However, it could be interpreted that for cities with low population densities, housing costs were likely around or above 61% (since below 61% would be for cities with negative population densities, which is impossible).
Choice A is incorrect because it interprets the values of the vertical axis as dollars and not percentages. Choice B is incorrect because the lowest housing cost is about 61% of the national average, not 61% of the highest housing cost. Choice C is incorrect because one cannot absolutely assert that no city with a low population density had housing costs below 61% of the national average, as the model shows that it is unlikely, but not impossible.
Let x and y be numbers such that −y < x < y. Which of the following must be true? I. x y < II. x > 0
III. y > 0

Solution
If −y < x < y, the value of x is either between −y and 0 or between 0 and y, so statement I, x < y is true. It is possible that the value of x is greater than zero, but x could be negative. For example, a counterexample to statement II, x > 0, is x = −2 and y = 3, yielding −3 < −2 < 3, so the given condition is satisfied. Statement III must be true since −y < x < y implies that −y < y, so y must be greater than 0. Therefore, statements I and III are the only statements that must be true. Choices A, B, and D are incorrect because each of these choices either omits a statement that must be true or includes a statement that could be false
f(x) = 2x^{3} + 6x^{2} + 4x
g(x) = x^{2} + 3x + 2
The polynomials f(x) and g(x) are defined above. Which of the following polynomials is divisible by 2x + 3?

Solution
In f(x), factoring out the greatest common factor, 2x, yields f(x) = 2x (x^{2} + 3x + 2). It is given that g(x) = x^{2} + 3x + 2, so using substitution, f(x) can be rewritten as f(x) = 2x ∙ g(x). In the equation p(x) = f(x) + 3g(x), substituting 2x ∙ g(x) for f(x) yields p(x) = 2x ∙ g(x) + 3 ∙ g(x). In p(x), factoring out the greatest common factor, g(x), yields p(x) = ( g(x))(2x + 3). Because 2x + 3 is a factor of p(x), it follows that p(x) is divisible by 2x + 3.
Choices A, C, and D are incorrect because 2x + 3 is not a factor of the polynomials h(x), r(x), or s(x). Using the substitution f(x) = 2x ∙ g(x), and factoring further, h(x), r(x), and s(x) can be rewritten as follows:
h(x) = (x + 1)(x + 2)(2x + 1)
r(x) = (x + 1)(x + 2)(4x + 3)
s(x) = 2 (x + 1)(x + 2)(3x + 1)
Because 2x + 3 is not a factor of h(x), r(x), or s(x), it follows that h(x), r(x), and s(x) are not divisible by 2x + 3.
In the circle above, segment AB is a diameter. If the length of arc \(\overset{\frown}{ADB}\) is 8π , what is the length of the radius of the circle?

Solution
Since segment AB is a diameter of the circle, it follows that arc \(\overset{\frown}{ADB}\) is a semicircle. Thus, the circumference of the circle is twice the length of arc \(\overset{\frown}{ADB}\) which is 2(8π) = 16π. Since the circumference of a circle is 2π times the radius of the circle, the radius of this circle is 16π divided by 2π, which is equal to 8.
Choices A, B, and D are incorrect and may be the result of losing track of factors of 2 or of solving for the diameter of the circle instead of the radius. For example, choice D is the diameter of the circle.
The tables below give the distribution of high temperatures in degrees Fahrenheit (°F) for City A and City B over the same 21 days in March.
Which of the following is true about the data shown for these 21 days?

Solution
The standard deviation is a measure of how far the data set values are from the mean. In the data set for City A, the large majority of the data are in three of the five possible values, which are the three values closest to the mean. In the data set for City B, the data are more spread out, with many values at the minimum and maximum values. Therefore, by observation, the data for City B have a larger standard deviation.
Alternatively, one can calculate the mean and visually inspect the difference between the data values and the mean. For City A the mean is \(\frac{1,655}{21}\) ≈ 78.8, and for City B the mean is \(\frac{1,637}{21}\) ≈ 78.0. The data for City A are closely clustered near 79, which indicates a small standard deviation. The data for City B are spread out away from 78, which indicates a larger standard deviation.
Choices A, C, and D are incorrect and may be the result of misconceptions about the standard deviation.
refer to the following information.
The bar graph above shows renewable energy consumption in quadrillions of British thermal units (Btu) in the United States, by energy source, for several energy sources in the years 2000 and 2010.
Of the following, which best approximates the percent decrease in consumption of wood power in the United States from 2000 to 2010 ?

Solution
Reading the graph, the amount of wood power used in 2000 was 2.25 quadrillion BTUs and the amount used in 2010 was 2.00 quadrillion BTUs. To find the percent decrease, find the difference between the two numbers, divide by the original value, and then multiply by 100:\(\frac{2.25 − 2.00}{2.25}\times100=\frac{0.25}{2.25}\times 100\simeq 11.1\)percent. Of the choices given,11% is closest to the percent decrease in the consumption of wood power from 2000 to 2010.
Choices A, C, and D are incorrect and may be the result of errors in reading the bar graph or in calculating the percent decrease.
refer to the following information.
The bar graph above shows renewable energy consumption in quadrillions of British thermal units (Btu) in the United States, by energy source, for several energy sources in the years 2000 and 2010.
In a scatterplot of this data, where renewable energy consumption in the year 2000 is plotted along the xaxis and renewable energy consumption in the year 2010 is plotted along the yaxis for each of the given energy sources, how many data points would be above the line y x = ?

Solution
The exact coordinates of the scatterplot in the xyplane cannot be read from the bar graph provided. However, for a data point to be above the line y = x, the value of y must be greater than the value of x. That is, the consumption in 2010 must be greater than the consumption in 2000. This occurs for 3 types of energy sources shown in the bar graph: biofuels, geothermal, and wind.
Choices A, B, and D are incorrect and may be the result of a conceptual error in presenting the data shown in a scatterplot. For example, choice B is incorrect because there are 2 data points in the scatterplot that lie below the line y = x.
The mass of living organisms in a lake is defined to be the biomass of the lake. If the biomass in a lake doubles each year, which of the following graphs could model the biomass in the lake as a function of time? (Note: In each graph below, O represents (0, 0).)

Solution
Since the biomass of the lake doubles each year, the biomass starts at a positive value and then increases exponentially over time. Of the graphs shown, only the graph in choice C is of an increasing exponential function.
Choice A is incorrect because the biomass of the lake must start at a positive value, not zero. Furthermore, this graph shows linear growth, not exponential growth. Choice B is incorrect because the biomass of the lake must start at a positive value, not zero. Furthermore, this graph has vertical segments and is not a function. Choice D is incorrect because the biomass of the lake does not remain the same over time.