SAT Practice Test 2 - SAT Math

A taxi charges a base fee plus $2.40 per mile. If the ride cost $18.60 for 5 miles, the equation is $x+2.40(5)=18.60$ , where $x$ is the base fee. What is $x$ ?
Answer: $6.60$ This problem asks for the base fee $x$ when a 5-mile taxi ride costs $18.60 total, with $2.40 per mile, modeled by $x + 2.40(5) = 18.60$ . First, calculate the mileage cost: $2.40(5) = 12.00$ . Then solve: $x + 12.00 = 18.60$ , so $x = 18.60 - 12.00 = 6.60$ . A common error would be to divide the total cost by the number of miles without accounting for the base fee. When solving cost problems with fixed and variable components, calculate the variable portion first.
The line shown passes through $(0,-3)$ and $(4,5)$ . What is the slope of the line?
Answer: $2$ To find the slope of a line passing through (0, -3) and (4, 5), we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁) = (5 - (-3))/(4 - 0) = (5 + 3)/4 = 8/4 = 2. The slope is 2, meaning the line rises 2 units for every 1 unit to the right. A common error is forgetting that subtracting a negative number means adding, so 5 - (-3) = 5 + 3, not 5 - 3. When one point has x = 0, it's also the y-intercept, making it easier to write the equation: y = 2x - 3.
A car's value, $V$ , depreciates by 18% each year. The value after $t$ years is modeled by $V=22000(0.82)^t$ . What does the number $22000$ represent in this situation?
Answer: Initial value at $t=0$ This question tests understanding of exponential decay model parameters, specifically what each number represents in context. In the equation V = 22000 $(0.82)^t$ , the number 22000 is the coefficient that appears when t = 0, making it the initial value of the car. The base 0.82 represents the decay factor (1 - 0.18 = 0.82, since the car loses 18% of its value each year). A common mistake is thinking 22000 represents the value after one year, but that would be 22000 $(0.82)^1$ = 18040. When interpreting exponential models, always evaluate at t = 0 to find the initial value.
A survey asked 200 students how they usually get to school. The pie chart shows the distribution among Bus, Car, Walk, Bike, and Other. How many more students chose Bus than Walk?
Answer: 40 students The question asks how many more students chose Bus than Walk out of 200 total students. From the pie chart, Bus shows 35% and Walk shows 15%. Converting to students: Bus = 200 × 0.35 = 70 students, Walk = 200 × 0.15 = 30 students. The difference is 70 - 30 = 40 students. A common error is subtracting percentages before converting to actual student counts.
A cyclist rides 18 miles in 1.5 hours at a constant speed. If the cyclist continues at the same speed, how long will it take to ride 30 miles?
Answer: 2.5 hours We need to find how long it takes to ride 30 miles at a constant speed. First, calculate the cyclist's speed: 18 miles ÷ 1.5 hours = 12 miles per hour. To find the time for 30 miles, divide distance by speed: 30 miles ÷ 12 miles/hour = 2.5 hours. The key is using distance/speed = time, not mixing up the rate formula. A common error is using hours per mile (1.5/18) instead of miles per hour. When dealing with rates, always check your units match what you're solving for.
A taxi fare is given by $F=3+2.2x$ , where $x$ is miles traveled. Which equation solves for $x$ in terms of $F$ ?
Answer: $x=\dfrac{F-3}{2.2}$ We need to solve $F = 3 + 2.2x$ for x in terms of F. First, subtract 3 from both sides: $F - 3 = 2.2x$ . Then divide both sides by 2.2: $x = \dfrac{F - 3}{2.2}$ . This formula allows us to find the distance traveled given the total fare. Common errors include adding 3 instead of subtracting, or inverting the fraction incorrectly. When isolating a variable, perform inverse operations in the reverse order of how the expression was built.
A phone battery has 100% charge at $t=0$ and then decreases by 12 percentage points every hour. Which equation models the battery percentage, $p$ , after $t$ hours (assuming the model continues)?
Answer: $p=100-12t$ A phone battery starts at 100% and decreases by 12 percentage points each hour. Since the decrease is constant (12 percentage points per hour), this is linear decay, not exponential. The equation is p = 100 - 12t, where we subtract 12t from the initial 100%. We can verify: at t=0, p=100%; at t=1, p=88%; at t=2, p=76%. The key distinction is "percentage points" (linear) versus "percent" (exponential) - losing 12 percentage points means going from 100% to 88%, not to 88% of 100%. Watch for this language carefully in word problems.
If $(x+4)(x-1)=x^2+bx-4$ , what is the value of $b$ ?
Answer: $3$ To find $b$ in $(x+4)(x-1)=x^2+bx-4$ , we expand the left side using FOIL. $(x+4)(x-1) = x^2 - x + 4x - 4 = x^2 + 3x - 4$ . Comparing with $x^2 + bx - 4$ , we see that $b = 3$ . The coefficient of $x$ comes from combining the outer and inner products: $(-1)x + 4x = 3x$ . Students often make sign errors in the FOIL process, especially with the inner and outer terms.
A movie theater sells adult tickets for $a$ dollars each and student tickets for $s$ dollars each. On Monday, $3$ adult tickets and $2$ student tickets cost a total of $54$ dollars. On Tuesday, $2$ adult tickets and $5$ student tickets cost a total of $80$ dollars. What is the value of $a+s$ ?
Answer: $22$ The two days give the system $3a + 2s = 54$ and $2a + 5s = 80$ . To eliminate $s$ , multiply the first equation by $5$ and the second by $2$ : $15a + 10s = 270$ and $4a + 10s = 160$ . Subtracting the second from the first gives $11a = 110$ , so $a = 10$ . Substituting into $3a + 2s = 54$ gives $30 + 2s = 54$ , so $2s = 24$ and $s = 12$ . Check with Tuesday: $2(10) + 5(12) = 20 + 60 = 80$ . So $a + s = 10 + 12 = 22$ . A value of $20$ is twice the adult price and $24$ is twice the student price, so both come from stopping after finding only one variable.
Two lines intersect forming four angles. One angle is labeled $(4x+18)^\circ$ and the adjacent angle (forming a linear pair) is labeled $(2x+24)^\circ$ . What is the value of $x$ ?
Answer: $23$ This question involves adjacent angles forming a linear pair. Linear pairs sum to $180^\circ$ , so $(4x + 18) + (2x + 24) = 180^\circ$ . Combining like terms: $6x + 42 = 180^\circ$ . Subtracting 42 from both sides: $6x = 138^\circ$ . Dividing by 6: $x = 23^\circ$ . To verify: when $x = 23$ , the angles are $4(23) + 18 = 110^\circ$ and $2(23) + 24 = 70^\circ$ , and $110^\circ + 70^\circ = 180^\circ$ . When working with linear pairs, always set their sum equal to $180^\circ$ .
A data set consists of the following values: 12, 15, 18, 22, 30, 35, 40. If the value 100 is added to the data set, which measure of center will be most affected?
Answer: Mean This question asks which measure of center is most affected by adding an extreme value (outlier). The mean is calculated by summing all values and dividing by the count, so adding 100 to the data set will significantly increase the sum and thus the mean. The median is the middle value when data is ordered, and adding one extreme value will have minimal impact on the middle position.
The temperature $T$ (in degrees) changes linearly with time $t$ (in hours). At $t=1$ , $T=68$ , and at $t=5$ , $T=60$ . Which equation models $T$ as a function of $t$ ?
Answer: $T=-2t+70$ Temperature T changes linearly with time t, with T = 68 when t = 1 and T = 60 when t = 5. The slope is m = (60 - 68)/(5 - 1) = -8/4 = -2 degrees per hour. Using point-slope form with (1, 68): T - 68 = -2(t - 1), which gives T = -2t + 2 + 68 = -2t + 70. The equation is T = -2t + 70. Common errors include mixing up which variable is dependent (T) versus independent (t), or sign errors in the slope calculation. In physics contexts, negative slope often indicates a decreasing quantity over time.
A gym charges a one-time registration fee of $15 plus $6 per visit. Maya has at most $75 to spend total. If $v$ is the number of visits, which inequality represents this situation, and what is the greatest integer value of $v$ that satisfies it?
Answer: $15+6v\le 75$ , greatest $v=10$ The question asks for the inequality modeling Maya's gym spending limit and the greatest integer number of visits v she can make. The total cost is a $15 fee plus $6 per visit, and she has at most $75, so the inequality is $15 + 6v \le 75$ . Subtract 15 from both sides to get $6v \le 60$ , then divide by 6 (positive, so no reversal) to find $v \le 10$ ; the greatest integer v is 10 since $15 + 6(10) = 75 \le 75$ . A common error is using a strict inequality $<$ instead of $\le$ , which would exclude $v=10$ even though it fits the budget. Another pitfall is subtracting incorrectly, like in choice C, leading to an overstated $v=15$ . When modeling real-world constraints like budgets, always check boundary values to confirm inclusion or exclusion.
A right triangle has hypotenuse length $13$ and one leg length $5$ . What is the area of the triangle?
Answer: $30$ Given a right triangle with hypotenuse 13 and one leg 5, we first find the other leg using the Pythagorean theorem. Setting up: 5² + b² = 13², we get 25 + b² = 169, so b² = 144, and b = 12. This is the classic 5-12-13 Pythagorean triple. The area of a right triangle equals (1/2) × base × height = (1/2) × 5 × 12 = 30. A common error is trying to use the hypotenuse in the area formula; remember that for right triangles, the area uses the two legs as base and height.
A student uses the equation $\frac{x-2}{5}=\frac{3x+1}{10}$ to compare two rates. What is the value of $x$ ?
Answer: $-5$ We need to solve $\frac{x-2}{5}=\frac{3x+1}{10}$ . Multiply both sides by 10: $\frac{10(x-2)}{5}=3x+1$ , which gives $2(x-2)=3x+1$ . Distribute: $2x-4=3x+1$ . Subtract $2x$ from both sides: $-4=x+1$ . Subtract 1: $-5=x$ . The key is to clear denominators by multiplying by the LCM (10), then solve the resulting linear equation. Be careful not to drop negative signs when moving terms.
A bacteria culture starts with 500 cells and doubles every 3 hours. Let $N(t)$ be the number of cells after $t$ hours. Which function best models $N(t)$ , and what is the growth factor per hour implied by the model?
Answer: $N(t)=500\cdot 2^{t/3}$ ; factor $2^{1/3}$ This problem describes bacteria that double every 3 hours, which is classic exponential growth with multiplicative change. For exponential growth, we use N(t) = N₀ · $r^t$ , where N₀ is the initial amount and r is the growth factor. Since the bacteria double every 3 hours, after 3 hours we have N(3) = 500 · 2 = 1000. To find the hourly growth factor, we need r³ = 2, so r = $2^{1/3}$ . Therefore, N(t) = 500 · $2^{t/3}$ , which can also be written as N(t) = 500 · $(2^{1/3})^t$ . The common mistake is confusing this with linear growth or incorrectly setting up the exponent. When dealing with doubling/tripling times, remember to divide the exponent by the time period.
How many solutions does the system have? $\begin{cases}4x+8y=16\\x+2y=5\end{cases}$ Decide whether the equations represent intersecting lines, parallel lines, or the same line.
Answer: $0$ We need to determine how many solutions the system $4x + 8y = 16$ and $x + 2y = 5$ has. Let me rewrite the first equation by dividing by 4: $x + 2y = 4$ . Now I have $x + 2y = 4$ and $x + 2y = 5$ . These equations have the same left side but different right sides ( $4 ≠ 5$ ), which means they represent parallel lines that never intersect. The tempting mistake mentioned is dividing the first equation by 4 and overlooking that $16/4 = 4$ , not 5. When two lines have the same slope but different y-intercepts, they are parallel and have zero solutions.
Simplify $\dfrac{x^2-9}{x-3}$ for $x \ne 3$ .
Answer: $x+3$ The question requires simplifying $(x^2 - 9)/(x - 3)$ for $x \ne 3$ by factoring and canceling. Factor the numerator as a difference of squares: $x^2 - 9 = (x - 3)(x + 3)$ . Thus, the expression is $(x - 3)(x + 3)/(x - 3)$ , and canceling the $(x - 3)$ factor gives $x + 3$ , provided $x \ne 3$ . This is the simplified form. A key error is canceling x terms without factoring, like subtracting $9/3 = 3$ to get $x - 3$ , or adding to $x + 3$ but incorrectly. Another mistake is ignoring the restriction and choosing something like $x^2 + 3$ . When simplifying rational expressions, factor completely before canceling and note domain restrictions.
A parabola has vertex at $(2,-3)$ and passes through the point $(0,5)$ . Which equation represents the parabola?
Answer: $y=2(x-2)^2-3$ A parabola with vertex $(2,-3)$ has the form $y=a(x-2)^2-3$ . To find $a$ , we use the fact that it passes through $(0,5)$ . Substituting: $5=a(0-2)^2-3=a(4)-3=4a-3$ . Solving: $5+3=4a$ , so $8=4a$ and $a=2$ . The equation is $y=2(x-2)^2-3$ . Common mistakes include using the wrong sign for $a$ or making arithmetic errors when solving for $a$ . Always verify your final equation by checking both the vertex and the given point.
A rectangular garden is $12\text{ m}$ long and $7\text{ m}$ wide. A straight walkway of uniform width $1\text{ m}$ runs along the inside of all four edges of the garden. What is the area of the remaining region available for planting, in square meters?
Answer: $50\text{ m}^2$ This problem asks for the area of the planting region after a 1-meter walkway is created inside a rectangular garden. The garden dimensions are 12 m × 7 m, and we need to find the area of the inner rectangle. Since the walkway is 1 m wide on all sides, the inner rectangle has dimensions (12 - 2×1) m × (7 - 2×1) m = 10 m × 5 m. The area of the planting region is 10 × 5 = 50 m². A common error is subtracting only 1 m from each dimension instead of 2 m (forgetting the walkway is on both sides).
A circle has equation $x^2+y^2+6x-8y=0$ . What is the radius of the circle?
Answer: $5$ This question requires finding the radius from a circle equation in general form. First, complete the square to convert to standard form. Starting with x² + y² + 6x - 8y = 0, group x and y terms: (x² + 6x) + (y² - 8y) = 0. Complete the square: (x² + 6x + 9) + (y² - 8y + 16) = 9 + 16, which gives (x+3)² + (y-4)² = 25. Therefore, r² = 25, so r = 5. A common mistake is forgetting to add the completing-the-square terms to both sides. When completing the square, always balance the equation.
The graph of a polynomial crosses the $x$ -axis at $x=-2$ and just touches (bounces off) the $x$ -axis at $x=3$ . Which factored form matches this behavior, assuming the leading coefficient is positive?
Answer: $(x+2)(x-3)^2$ We need to identify which factored form matches a polynomial that crosses at $x=-2$ and touches (bounces) at $x=3$ . When a polynomial crosses the x-axis at a zero, that factor has odd multiplicity (usually 1); when it touches and bounces, the factor has even multiplicity (usually 2). Since it crosses at $x=-2$ , we need factor $(x+2)$ with odd power; since it touches at $x=3$ , we need factor $(x-3)$ with even power. With positive leading coefficient, the polynomial is $(x+2)(x-3)^2$ . The key insight is that multiplicity determines crossing versus touching behavior. Remember: odd multiplicity means the graph crosses through; even multiplicity means it touches and turns around.
A streaming service charges a one-time setup fee plus a monthly fee. The total cost after 6 months is modeled by $18 + 6x = 72$ , where $x$ is the monthly fee in dollars. What is the value of $x$ ?
Answer: $9$ This problem asks us to find the monthly fee $x$ given that the total cost after 6 months is $72. Starting with $18 + 6x = 72$ , we first subtract 18 from both sides: $6x = 72 - 18 = 54$ . Then we divide both sides by 6: $x = 54 \div 6 = 9$ . The key error to avoid is dividing 72 by 6 first (getting 12) before subtracting the setup fee, which would give the wrong answer. When solving equations, always isolate the variable term before dividing.
On a coordinate plane, a line $\ell$ passes through the points $(-4,5)$ and $(2,-1)$ . What is the slope of $\ell$ ?
Answer: $-1$ To find the slope of line $\ell$ passing through $(-4,5)$ and $(2,-1)$ , we use the slope formula. Calculate: $m = \frac{-1-5}{2-(-4)} = \frac{-6}{6} = -1$ . The negative slope indicates the line decreases from left to right. Students often make sign errors when subtracting negative x-coordinates - remember that $2-(-4) = 2+4 = 6$ . When finding slope, always double-check by verifying that rise over run gives the same result regardless of which point you label as $(x_1,y_1)$ .
A shipping box has a mass of $2.4\text{ kg}$ . Convert this mass to grams using $1\text{ kg}=1{,}000\text{ g}$ . Which answer is correct?
Answer: 2,400 g We need to convert 2.4 kg to grams using 1 kg = 1,000 g. Set up the conversion: 2.4 kg × (1,000 g / 1 kg). Multiply: 2.4 × 1,000 = 2,400 g, with kg units canceling out. A common error is misplacing the decimal point when multiplying by 1,000. Remember that grams are smaller than kilograms, so we expect a larger numerical value when converting kg to g.
A water tank contains 50 liters initially and is being filled at a constant rate of 6 liters per minute. Let $t$ be time in minutes and $V$ be volume in liters. For what value of $t$ will $V=110$ ?
Answer: $10$ The water tank problem gives us initial volume 50 liters and filling rate 6 liters/minute, so $V = 50 + 6t$ . We need to find when $V = 110$ . Setting up the equation: $110 = 50 + 6t$ , which gives us $60 = 6t$ , so $t = 10$ minutes. Common errors include forgetting to add the initial amount or setting up the equation incorrectly. When solving word problems, always define your variables clearly and check that your answer makes sense in context.
A bar graph shows the number of visitors to a museum on Saturday. According to the graph, what is the median number of visitors among these five hours?
Answer: 55 visitors The question asks for the median number of visitors among five hourly counts: 40, 55, 70, 65, 50. To find the median, we arrange these in order: 40, 50, 55, 65, 70. The median is the middle value, which is 55 visitors. A common error would be calculating the mean instead of the median or not properly ordering the values first.
A class has $28$ students. The ratio of students who brought lunch to students who bought lunch is $9:5$ . How many students bought lunch?
Answer: 10 This problem asks how many students bought lunch when 28 students total have a brought:bought ratio of 9:5. The ratio tells us that for every 9 + 5 = 14 students total, 5 bought lunch. To find how many bought lunch: (5/14) × 28 = 5 × 2 = 10 students. The key is recognizing that the ratio 9:5 means the class is divided into 14 equal parts. Always add ratio parts to find what fraction each group represents of the total.
A population is modeled by $P(t)=1200(0.85)^t$ , where $t$ is measured in years. Which statement best interprets the number 0.85 in the model?
Answer: Population keeps 85% each year This question asks us to interpret the base 0.85 in the exponential model P(t) = 1200 $(0.85)^t$ . In exponential models of the form y = $a(b)^t$ , the base b represents the growth/decay factor - what the quantity is multiplied by each time period. Since 0.85 < 1, this represents decay, and 0.85 means the population retains 85% of its value each year (or loses 15%). The population doesn't decrease BY 0.85 or BY 85%; rather, it's multiplied by 0.85. Understanding the difference between "decreases by" (subtraction) and "decreases to" (multiplication) is crucial for interpreting exponential models.
On the coordinate plane, the line passes through the points $(0,4)$ and $(2,0)$ . What is the value of $y$ when $x=5$ on this line?
Answer: $-6$ Given points (0,4) and (2,0), we first find the slope: m = (0-4)/(2-0) = -4/2 = -2. The y-intercept is 4 (from point (0,4)), so the equation is y = -2x + 4. When x = 5: y = -2(5) + 4 = -10 + 4 = -6. The negative slope indicates y decreases as x increases. Common errors include sign mistakes when calculating slope or substituting x = 5. Always verify your equation works for both given points before using it for predictions.
Lines $\ell$ and $m$ are parallel and cut by a transversal. One interior angle on the same side of the transversal measures $68^\circ$ . What is the measure of the other same-side interior angle?
Answer: $112^\circ$ This question asks for the measure of a same-side interior angle when parallel lines are cut by a transversal. The key property is that same-side interior angles (also called co-interior angles) are supplementary, meaning they add up to $180^\circ$ . Since one angle measures $68^\circ$ , the other same-side interior angle measures $180^\circ - 68^\circ = 112^\circ$ . A common mistake is confusing same-side interior angles with alternate interior angles, which are equal rather than supplementary. When dealing with parallel lines and transversals, identify the angle relationship first before applying the appropriate property.
A cross-country runner recorded the number of miles run each day for 9 days: 3, 4, 4, 5, 5, 5, 6, 6, 20. The coach wants a single number to represent a "typical" day of running. Which measure of center is most affected by the outlier value 20, and what is the median of the data set? Choose the option that correctly identifies both.
Answer: Mean; median $=5$ This question asks which measure of center is most affected by the outlier (20) and what the median is. To find the median of 9 values, we order them: 3, 4, 4, 5, 5, 5, 6, 6, 20, then take the 5th value (middle position), which is 5. The mean is (3+4+4+5+5+5+6+6+20)÷9 = 58÷9 ≈ 6.4, while without the outlier it would be 38÷8 = 4.75, showing the mean changes dramatically. The median remains 5 whether we include the outlier or not, demonstrating that the mean is most affected by outliers. When dealing with outliers, always check how each measure of center responds to extreme values.
A baker mixes flour and sugar in a ratio modeled by $\frac{x-2}{3}=\frac{5}{6}.$ If $x$ represents the number of cups of flour used, what is the value of $x$ ?
Answer: $\frac{9}{2}$ We need to solve $\frac{x-2}{3} = \frac{5}{6}$ for the number of cups of flour $x$ . Cross-multiplying gives us $6(x-2) = 3(5)$ , which simplifies to $6x - 12 = 15$ . Adding 12 to both sides: $6x = 27$ , then dividing by 6: $x = \frac{27}{6} = \frac{9}{2} = 4.5$ . A common error is to incorrectly cross-multiply or to make arithmetic mistakes when simplifying the fraction. When solving proportions, cross-multiplication provides a direct path to the solution.
The value of a used car decreases linearly. It is 17000 dollars after 2 years and 11000 dollars after 5 years. Let $V(t)$ be the value in dollars after $t$ years since purchase. Which equation models $V(t)$ ?
Answer: $V(t)=-2000t+21000$ Slope is $(11000-17000)/(5-2)=-2000$ and $b$ satisfies $17000=-2000(2)+b$ , so $b=21000$ . Other options have the wrong slope or intercept.
Consider the system: $\begin{cases} 4x-6y=12\\ 2x-3y=6 \end{cases}$ How many solutions does the system have?
Answer: infinitely many We need to determine how many solutions this system has. Looking at the equations $4x - 6y = 12$ and $2x - 3y = 6$ , I notice that if we multiply the second equation by 2, we get $4x - 6y = 12$ . This is exactly the same as the first equation! When two equations in a system are identical (or one is a multiple of the other), they represent the same line. Since both equations describe the same line, every point on that line is a solution. The common error here is to try solving without recognizing this relationship, which might lead you to think there's exactly one solution. When working with systems, always check if one equation is a scalar multiple of the other before attempting elimination or substitution.
Factor the trinomial $6x^2+11x+3$ . Choose the correct factorization. Several choices seem
Answer: $(3x+1)(2x+3)$ To factor $6x^2+11x+3$ , we need two binomials whose product gives this trinomial. The first terms must multiply to $6x^2$ (options: $6x cdot x$ or $3x cdot 2x$ ) and the last terms must multiply to $3$ (options: $3 cdot 1$ ). Testing $(3x+1)(2x+3)$ : First terms give $6x^2$ , outer gives $9x$ , inner gives $2x$ , last gives $3$ . Middle term: $9x+2x = 11x$ ✓. Therefore, $6x^2+11x+3 = (3x+1)(2x+3)$ . A common error is pairing factors incorrectly, like $(6x+1)(x+3)$ which gives a middle term of $19x$ . When factoring trinomials with leading coefficient ≠ 1, systematically test factor pairs.
A parabola on the coordinate plane has vertex at $(-2,3)$ and opens upward. It passes through the point $(0,11)$ . Which equation represents the parabola?
Answer: $y=2(x+2)^2+3$ The question asks for the equation of an upward-opening parabola with vertex (-2, 3) that passes through (0, 11). Vertex form is y = $a(x - h)^2$ + k, and h = -2 makes the factor (x + 2), so y = $a(x + 2)^2$ + 3. Substituting the given point: 11 = $a(0 + 2)^2$ + 3 = 4a + 3, so 4a = 8 and a = 2, giving y = 2 $(x + 2)^2$ + 3. The equation with $(x + 2)^2$ and no coefficient uses a = 1 and therefore passes through (0, 7) instead of (0, 11). Both equations built on $(x - 2)^2$ place the vertex at x = 2 rather than x = -2, flipping the sign of the horizontal shift; the one without a coefficient misses the vertex and the point.
On the coordinate plane, vertices of a triangle are at (1, 2), (4, 6), and (7, 2). What is the area of the triangle?
Answer: 12 units² This question asks for the area of a triangle with vertices at (1, 2), (4, 6), and (7, 2). Notice that two vertices, (1, 2) and (7, 2), have the same y-coordinate, forming a horizontal base of length 7 - 1 = 6 units. The height is the perpendicular distance from (4, 6) to this base, which is 6 - 2 = 4 units. Using A = ½ × base × height: A = ½ × 6 × 4 = 12 square units.
In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle, the hypotenuse is $18$ . What is the length of the longer leg?
Answer: $9\sqrt{3}$ In a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle with side ratio $1:\sqrt{3}:2$ , if the hypotenuse is 18, we can find all sides. The shortest side (opposite $30^\circ$ ) $= 18/2 = 9$ , and the longer leg (opposite $60^\circ$ ) $= 9\sqrt{3}$ . The longer leg is always the side opposite the $60^\circ$ angle and equals the short side times $\sqrt{3}$ . Common errors include confusing which leg is longer or using the wrong special triangle ratios. The answer $9\sqrt{3}$ follows the standard $30$ - $60$ - $90$ pattern.
A company's revenue $R$ (in thousands of dollars) is modeled by $R(t)=50+4t$ , where $t$ is years since 2018. Another analyst claims the model should be exponential: $R(t)=50(1.08)^t$ . If both models match at $t=0$ , which model predicts a larger revenue at $t=10$ ?
Answer: Exponential model We need to compare R(t) = 50 + 4t (linear) with R(t) = 50 $(1.08)^t$ (exponential) at t = 10. For the linear model: R(10) = 50 + 4(10) = 90. For the exponential model: R(10) = 50 $(1.08)^{10}$ ≈ 50(2.159) ≈ 107.9. Since 107.9 > 90, the exponential model predicts larger revenue. This illustrates a key principle: exponential growth eventually overtakes linear growth, even if the linear model grows faster initially. Both models equal 50 at t = 0 as stated.
A circle has a central angle of $120^\circ$ that subtends an arc of length $10\pi$ inches. What is the radius of the circle, in inches?
Answer: $15$ The question asks for the radius of a circle given a central angle and arc length. The formula relating arc length, radius, and central angle (in radians) is $s = r\theta$ , where $s$ is arc length, $r$ is radius, and $\theta$ is the angle in radians. First, convert $120°$ to radians: $120° = 120 \cdot \frac{\pi}{180} = \frac{2\pi}{3}$ radians. Now substitute into the formula: $10\pi = r \cdot \frac{2\pi}{3}$ , so $r = \frac{10\pi}{\frac{2\pi}{3}} = 10\pi \cdot \frac{3}{2\pi} = 15$ inches. A common error is forgetting to convert degrees to radians before using the arc length formula. Always check whether angles are given in degrees or radians.
A boat travels due east across a river that is 300 m wide. The boat's path makes a $15^\circ$ angle south of east because of the current. Approximately how far does the boat travel from one bank to the other (nearest meter)?
Answer: 311 m A boat travels across a 300 m wide river, but its path makes a $15^\circ$ angle south of east due to current. The 300 m width represents the side adjacent to the $15^\circ$ angle, and the actual distance traveled is the hypotenuse, so we use cosine: $\cos(15^\circ) = 300/\text{distance}$ . Solving: $\text{distance} = 300/\cos(15^\circ) = 300/0.966 \approx 310.6$ m, which rounds to 311 m. A common error is treating the 300 m as the hypotenuse instead of the adjacent side. In navigation problems, carefully identify which measurement corresponds to which side of the triangle.
A drawer contains 7 black socks and 5 white socks. Two socks are drawn at random without replacement. What is the probability that the two socks are different colors?
Answer: $\frac{35}{66}$ This asks for the probability of drawing two socks of different colors from 7 black and 5 white socks (12 total) without replacement. We can get different colors in two ways: black then white, or white then black. P(black then white) = $\frac{7}{12} \times \frac{5}{11} = \frac{35}{132}$ . P(white then black) = $\frac{5}{12} \times \frac{7}{11} = \frac{35}{132}$ . Total probability = $\frac{35}{132} + \frac{35}{132} = \frac{70}{132} = \frac{35}{66}$ . A common error is forgetting one of the two orderings. When order matters for achieving an outcome, consider all possible sequences.
Solve the inequality $\frac{2-5x}{-3}\ge 4$ . What is the solution set?
Answer: $x\ge \frac{14}{5}$ We need to solve (2 - 5x)/(-3) ≥ 4. First, multiply both sides by -3, and since we're multiplying by a negative number, we MUST reverse the inequality sign: 2 - 5x ≤ -12. Next, subtract 2 from both sides: -5x ≤ -14. Finally, divide by -5, again reversing the inequality sign: x ≥ 14/5. The solution involves two sign reversals because we work with negative numbers twice. This problem tests whether you remember to flip the inequality sign EACH time you multiply or divide by a negative.

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