When calculating a certain tax, Simone found that no tax is due for all income up to p dollars, with p > 0. Income greater than p dollars is taxed at a rate of 15%. Which of the following graphs accurately represents this tax?

Solution

For income less than p dollars, the tax rate is $0, so the line should begin at the origin; therefore, you can rule out Choices (A) and (E).

For income greater than p dollars, the tax rate increases at a steady rate without leveling off, so you can rule out Choices (C) and (D).

By elimination you know that the correct answer is Choice (B).

Doug, who runs track for his high school, was challenged to a race by his younger brother, Matt. Matt started running first, and Doug didn’t start running until Matt had finished a quarter-mile lap on the school track. Doug passed Matt as they both finished their sixth lap. If both boys ran at a constant speed, with Doug running 2 miles an hour faster than Matt, what was Matt’s speed?

10.5 miles per hour
10 miles per hour
9 miles per hour
8 miles per hour
7.5 miles per hour

Solution

Doug runs 2 miles an hour faster than Matt, so let Matt’s speed equal x miles per hour.

Then Doug’s speed equals x + 2 miles per hour.

Each lap is one-quarter of a mile, so Doug runs 1.5 miles in the time it takes Matt to run 1.25 miles.

Place this information in a chart:

Rate Time Distance

Doug x + 2 $\frac{1.5}{x + 2}$ 1.5

Matt x $\frac{1.25}{x}$ 1.25

The two boys took the same amount of time from the time Doug started, so make an equation by setting the two times in the chart equal to each other, and then solve for x:

$\frac{1.5}{x + 2} = \frac{1.25}{x}$

1.5x = 1.25(x + 2)

1.5x = 1.25x + 2.5

0.25x = 2.5

x = 10

So Matt ran at 10 miles per hour.

On an xy-graph, three corners of a parallelogram are located at (3, 3), (4, – 4), and (–2, –1). Which of the following points could be the remaining corner?

(8, 0)
(8, –1)
(–1, 9)
(–3, 6)
(–5, 7)

Solution

A good way to begin is to draw a picture showing the three points given, including possible places where a fourth point would form a parallelogram.

Here’s what your picture may look like:

This figure shows three possible points for the remaining corner of the parallelogram: A, B, and C.

To find the exact coordinates of these three additional points, choose any of the given points and count up and over (or down and over) to a second given point.

Then, starting from the third given point, count the same number of steps up and over (or down and over) and label the point where you end up.

By this method, you find that A = (–3, 6), B = (9, 0), and C = (–1, –8).

Only point A is listed as an answer.

If a + 2b = 2, what is the value of $\left ( \frac{a}{b – 1} \right ) + \left ( \frac{a}{b – 1} \right )^{2} + \left ( \frac{a}{b – 1} \right )^{3}$?

-6
8
10
-12
14

Solution

Begin by finding the value of $\frac{a}{b - 1}$:

a + 2b = 2

a = 2 - 2b

a = 2(1 - b)

a = -2(b - 1)

$\frac{a}{b - 1} = -2$

Now substitute –2 for $\frac{a}{b - 1} = -2$ to get your answer:

$\left ( \frac{a}{b - 1} \right ) + \left ( \frac{a}{b - 1} \right )^{2} + \left ( \frac{a}{b - 1} \right )^{3}$

= -2 + (-2)² + (-2)³

= -2 + 4 - 8

= -6

If you plot the equation x² + (y – 2)² = 4 as a circle on a standard xy-graph, what is the area of the circle’s region that will lie in Quadrant 1, as shown in the following figure?

Zero
π
2π
4π
16π

Solution

The formula for a circle with a radius r centered at (a , b) is (x – a)² + (y – b)² = r².

Thus, x² + (y – 2)² = 4 has a radius of 2 and is centered at (0, 2), as shown here:

This circle has a radius of 2, so calculate its total area as follows:

A = πr² = π(2)² = 4π

Because half the circle is in Quadrant 1, the area of this region is 2π.

If x² – x – 2 > 0, which of the following is the solution set for x?

x > –1
x > 2
–1 < x < 2
x < –1 or x > 2
No solutions are possible.

Solution

Begin by treating the inequality x² – x – 2 > 0 as if it were an equation.

Factor to find the zeros:

x² – x – 2 = 0

(x + 1)(x – 2) = 0

x = –1 and x = 2

The graph of this function is a parabola that crosses the x-axis at –1 and 2.

This graph is concave up because the coefficient of the x² term (that is, a) is positive.

The parabola dips below the x-axis when –1 < x < 2, so these values do not satisfy the equation. Therefore, the correct answer is Choice (D). (If you doubt this answer, note that when x = 0, x² – x – 2 = –2, which is less than 0. So 0 is not in the solution set.)

The following graph shows the number of new bank accounts that eight account executives have opened so far this week. Which of the following answer choices is the median number of accounts opened among these eight people?

9.5
10
10.5
11
12

Solution

Place the numbers of accounts sold in order from smallest to largest:

7 8 9 10 11 12 12 14

The middle two numbers are 10 and 11, so the median is the average of these two numbers, which is 10.5.

What is the area of the trapezoid in the following figure?

7
7√2
7√3
$\frac{7\sqrt{3}}{2}$
$\frac{7\sqrt{3}}{4}$

Solution

To begin, add a line to the figure, cutting off a 30-60-90 triangle on the left side of the trapezoid, as shown here:

Notice that the short leg of this triangle has a length of 1, so the height of the trapezoid is √3.

Plug this value, plus the lengths of the two bases, into the formula for a trapezoid to get your answer:

$A = \frac{b_{1} + b_{2}}{2}h = \frac{3 + 4}{2}\sqrt{3} = \frac{7\sqrt{3}}{2}$

If $\begin{vmatrix} 6 & x & y \end{vmatrix} + \begin{vmatrix} x & y & z \end{vmatrix} = \begin{vmatrix} 9 & 7 & 5 \end{vmatrix}$, then x + y + z =

Solution

To add two matrices, you add the corresponding components.

So this matrix equation yields the following three equations:

6 + x = 9

x + y = 7

y + z = 5

In the first equation, x = 3.

Substituting 3 for x in the second equation gives you 3 + y = 7, so y = 4.

If you substitute 4 for yin the third equation, you get 4 + z = 5; therefore, z = 1.

Therefore, x + y + z = 3 + 4 + 1 = 8.

In the following figure, the line passes through the origin and through P = (1, 3). If you were to draw a new line perpendicular to the first that also passes through P,what would be the equation of this new line?

y = 3x + 4
y = ¹⁄₃x + 4
y = ¹⁄₃x + ¹⁰⁄₃
y = -¹⁄₃x + 4
y = -¹⁄₃x + ¹⁰⁄₃

Solution

The line that’s shown in the figure passes through (0, 0) and (1, 3), so its slope is “up 3, over 1”: ³⁄₁ = 3.

Thus, the new line that’s perpendicular to this line, and passing through P= (1, 3), will have a negative reciprocal slope, which is -¹⁄₃.

To find the equation of this line, plug these numbers into the slope-intercept form to get b:

y = mx + b

3 = -¹⁄₃(1) + b

3 = --¹⁄₃ + b

¹⁰⁄₃ = b

Finally, plug in the values to get the equation for this line:

y = -¹⁄₃x + ¹⁰⁄₃

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	Rate	Time	Distance
Doug	x + 2	\(\frac{1.5}{x + 2}\)	1.5
Matt	x	\(\frac{1.25}{x}\)	1.25