The graph of f(x) is shown above in the xy-plane. The points (0, 3), (5b, b), and (10b, -b) are on the line described by f(x). If b is a positive constant, what are the coordinates of point C ?
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Solution
Right away, (A) can be eliminated, since point C has a negative y-coordinate.
Given any two points, the slope of the line can be determined using the equation \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\).
Use this formula to find the value of b by setting the slope of \(\overline{AB}\) equal to the slope of \(\overline{BC}\).
Use points (0, 3) and (5b, b) in the left side of the equation and points (5b, b) and (10b, -b) in the right side of the equation to get \(\frac{3 - b}{0 - 5b} = \frac{-b - b}{10b - 5b}\).
Simplify both sides of the equation to get \(\frac{3 - b}{-5b} = \frac{-2b}{5b} \;\; or \;\; \frac{3 - b}{-5b} = \frac{-2}{5}\).
Cross-multiply to get 5(3 - b) = 10b.
Divide both sides by 5 get 3 - b = 2b, then 3 = 3b, and finally b = 1.
Plug in b =1 for point C to get [10(1), - (1)], or (10, -1).
Therefore, the correct answer is (B).
(x – 2)2 + y2 = 36
y = -x + 2
The equations above represent a circle and a line that intersects the circle across its diameter. What is the point of intersection of the two equations that lies in quadrant II?
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Solution
In quadrant II, the x-coordinate is negative, and the y-coordinate is positive.
Therefore, eliminate (C).
Whenever the question includes variables and the answers are numbers, think Plugging In the Answers.
Of the remaining answers, (B) is easiest to work with.
In (B), the x-value is -4 and the y-value is 2.
Plug these values into the second equation to get -4 = -2 + 2.
Given that this is not a true statement, eliminate (B).
Try the values in (A) in the second equation to get 3√2 = -(-3√2) + 2.
This is also not true, so the correct answer is (D).
United States Investment in
Alternative Energy Sources
| Actual 2007 Investment | Projected 2017 Investment | |
| Biofuels | 0.31 | 0.34 |
| Wind | 0.40 | 0.32 |
| Solar | 0.27 | 0.30 |
| Fuel Cells | 0.02 | 0.04 |
| Total | 1.00 | 1.00 |
The table above shows the relative investment in alternative energy sources in the United States by type. One column shows the relative investment in 2007 of $75 million total invested in alternative energy. The other column shows the projected relative investment in 2017 given current trends. The total projected investment in alternative energy in 2017 is $254 million. Suppose that a new source of alternative energy, Cold Fusion, is perfected. It is projected that by 2017 that $57 million will be invested in Cold Fusion in the United States, without any corresponding reduction in investment for any other form of alternative energy. What portion of the total investment of alternative energy in the United States will be spent on biofuels?
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Solution
First, you know the new proportion must be less than the current 0.34 for biofuels (because the total amount spent on alternative energy is increasing, but the amount spent on biofuels is remaining the same), so you can eliminate (D).
Next, determine the amount that will be spent on biofuels in 2017 by multiplying 0.34 by the total of $254 million: 0.34 × 254 = $86.36 million.
Because 57 million new dollars will be spent on alternative energy, the new total will be 254 + 57 = $311 million.
Divide $86.36 million by $311 million to get the new proportion: \(\frac{86.38}{311} = 0.28\), which is (C).
-24 – 8j = 12k
3 + 5⁄3k = –7⁄6j
Which of the following ordered pairs (j, k) is the solution to the system of equations above?
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Solution
Whenever there are variables in the question and numbers in the answer choices, think Plugging In the Answers.
In (A), j = 6, and k = -6. Plug these two values into the first equation to get -24 - 8(6) = 12(-6).
Solve for both sides of the equation to get -24 - 48 = -72, or -72 = -72.
Therefore, the values work for the first equation.
Plug the values into the second equation to get 3 + 5⁄3(-6) = -7⁄6(6).
Solve both sides of the equation to get 3 + (-10) = -7, or -7 = -7.
Since the values given in (A) work in both equations, the correct answer is (A).
If –20⁄7 < -3z + 6 < –11⁄5, what is the greatest possible integer value of 9z – 18 ?
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Solution
When solving inequalities, the natural impulse is to isolate the variable.
In this case, though, look at what the question is asking. The question doesn't want you to find just the the value of z but rather the value of 9z - 18.
To get from the value of -3z + 6 given in the inequality to this new value, the original inequality must be multiplied by -3.
Just multiply the entire inequality by this value, making sure to flip the inequality signs when multiplying by a negative number.
The equation becomes -3(-20⁄7) > -3(-3z + 6) > -3(-11⁄5) or 60⁄7 > 9z - 18 > 33⁄5.
The question asks for the greatest possible integer value, so focus on the high end of the given values.
The value at that end, 60⁄7 equals 8.57, so the greatest integer less than that is 8. The answer is (C).
In a certain sporting goods manufacturing company, a quality control expert tests a randomly selected group of 1,000 tennis balls in order to determine how many contain defects. If this quality control expert discovered that 13 of the randomly selected tennis balls were defective, which of the following inferences would be most supported?
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Solution
The quality control expert discovered that 13 out of 1,000 randomly selected tennis balls were defective.
\(\frac{13}{1000} = 0.013\), which is equivalent to 1.3%.
This means that 100 - 1.3 = 98.7% of tennis balls tested were not defective, and this data most supports answer (B).
Lennon has 6 hours to spend in Ha Ha Tonka State Park. He plans to drive around the park at an average speed of 20 miles per hour, looking for a good trail to hike. Once he finds a trail he likes, he will spend the remainder of his time hiking it. He hopes to travel more than 60 miles total while in the park. If he hikes at an average speed of 1.5 miles per hour, which of the following systems of inequalities can be solved for the number of hours Lennon spends driving, d, and the number of hours he spends hiking, h, while he is at the park?
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Solution
Start with the easiest piece of information first, and use Process of Elimination.
Given that h is the number of hours spent hiking and d is the number of hours driving, the total number of hours Lennon spends in the park can be calculated as h + d.
The question states that Lennon has up to 6 hours to spend in the park-"up to" means ≤. So, h + d ≤ 6.
Eliminate (B), (C), and (D).
The correct answer is (A).
If \(\frac{\left(C + x \right)}{x – 3} = \frac{x + 8}{3}\), which of the following could be an expression of C in terms of x ?
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Solution
Cross-multiply to get 3(C + x) = (x - 3)(x + 8).
Expand the right side of the equation to get 3(C + x) = x2 + 5x - 24.
Distribute the 3 to get 3C + 3x = x2 + 5x - 24.
Subtract 3x from both sides of the equation to get 3C = x2 + 2x - 24.
Factor the right side of the equation to get 3C = (x + 6)(x - 4).
Divide both sides by 3 to get C = \(\frac{\left(x + 6 \right)\left(x - 4 \right)}{3} = \frac{1}{3}\left(x + 6 \right)\left(x - 4 \right)\).
The correct answer is (C).
Bryan, who works in a high-end jewelry store, earns a base pay of $10.00 per hour plus a certain percent commission on the sales that he helps to broker in the store. Bryan worked an average of 35 hours per week over the past two weeks and helped to broker sales of $5,000.00 worth of jewelry during that same two-week period. If Bryan’s earnings for the two-week period were $850.00, what percent commission on sales does Bryan earn?
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Solution
There are a few different ways to approach this question.
In any approach, the best first step is to figure out how much income Bryan earned during the two-week period without the commission.
Since he worked an average of 35 hours per week for two weeks, he worked a total of 70 hours.
At a rate of $10.00 per hour base pay, this would add up to $700.00 (70 × 10 = 700).
Since Bryan's earnings were actually $850.00, that means he must have earned $150.00 of commission (850 - 700 = 150).
At this point, you can calculate the percent commission algebraically or simply work backwards from the answers.
Algebraically, you know that $150.00 is equal to a certain percent of $5,000.00 in sales, which can be represented as follows: \(150 = \frac{x}{100}\left ( 5,000 \right )\).
Solve for x, and you get 3, which is (C).
If instead you wish to work backwards from the answers, you can take the answers and calculate what 1%, 2%, etc. of $5,000.00 would be, and then add that back to $700.00 to see which choice matches your target of $850.00: (C).
Bailey’s Boutique Clothing is having a 20% off sale during which shirts cost $30.00 and pants cost $60.00. On the day of the sale, Bailey’s sells 2⁄3 a total of 60 shirts and pants and earned a total of $2,250. On a regular day, Bailey’s sells the number of shirts and pants sold during the sale and earns a total of $1,875. Solving which of the following systems of equations yields the number of shirts, s, and the number of pants, p, sold during a regular day?
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Solution
Start with the easier equation and use Process of Elimination.
The easier equation is related to the total number of shirts and pants, s + p, sold on a regular day.
The question states that on a regular day Bailey's sells 2⁄3 the number of pants and shirts sold during a sale. 2⁄3(60) = 40.
Therefore, one of the equations in the correct answer should be s + p = 40.
Eliminate (C) and (D) since neither of these answers include this equation.
The other equation is related to the money Bailey's earns on a regular day.
According to the question, Bailey's earns a total of $1,875 on a regular day, so the equation must equal $1,875.
Eliminate (B) because the total in the money equation is incorrect.
The correct answer is (A).