SAT Practice Test 7 - SAT Math

A physics relationship is $F = ma$ . Solve for $a$ in terms of $F$ and $m$ .
Answer: $a=\frac{F}{m}$ We need to solve $F = ma$ for acceleration $a$ . Since $a$ is multiplied by $m$ , we divide both sides by $m$ to isolate $a$ : $\frac{F}{m} = \frac{ma}{m} = a$ . Therefore, $a = \frac{F}{m}$ . The common mistake is dividing by the wrong variable (like dividing by $a$ instead) or solving for the wrong variable (solving for $m$ instead of $a$ ). When solving for a specific variable, identify what's multiplied with it and divide both sides by those factors.
A histogram shows how many items each customer had in their shopping cart in a single day at a grocery store. Which range of items-per-cart was the most common among shoppers that day?
Answer: 20–29 items The question asks for the modal bin (most common range) in the histogram. Examining the bar heights for each range, the 20-29 items bin has the highest frequency. This is the mode because it represents the interval with the most occurrences. Students sometimes confuse mode with mean or median, or misread which bar is tallest.
A line has $y$ -intercept 5 and passes through the point $(2,1)$ . Which equation represents the line?
Answer: $y=-2x+5$ The question asks for the equation of a line with y-intercept 5 passing through (2,1). Find slope m=(1-5)/(2-0)=-4/2=-2. Thus, y=-2x+5. This models the line accurately. A key error is using (2,1) as intercept, miscounting slope as 1/2. Another mistake could be positive slope calculation. When given intercept and point, compute slope then write in slope-intercept form.
A card is drawn at random from a standard 52-card deck. What is the probability that the card is a heart and a face card (jack, queen, or king)?
Answer: $\tfrac{3}{52}$ This question asks for the probability of drawing a card that is both a heart AND a face card from a standard 52-card deck. A standard deck has 13 hearts and 12 face cards total (3 per suit: J, Q, K), but we need cards that satisfy both conditions. The heart suit contains exactly 3 face cards: jack of hearts, queen of hearts, and king of hearts. Therefore, $P(\text{heart AND face card}) = \tfrac{3}{52}$ . A common error is multiplying $P(\text{heart}) \times P(\text{face card}) = \tfrac{13}{52} \times \tfrac{12}{52}$ , which assumes independence and gives the wrong answer. For AND probabilities, count only the outcomes that satisfy both conditions simultaneously.
A bakery makes $x$ batches of cookies. Each batch uses 3 cups of flour, and the bakery has 20 cups available. They must also make at least 4 batches. Which compound inequality represents all possible values of $x$ ?
Answer: $4\le x\le \tfrac{20}{3}$ This question requires a compound inequality for x batches of cookies, each using 3 cups with 20 available, and at least 4 batches. For flour: $3x \leq 20$ , $x \leq \frac{20}{3}$ ; for minimum: $x \geq 4$ . Combined: $4 \leq x \leq \frac{20}{3}$ . A key error is using strict inequalities, excluding boundaries. Another mistake is reversing the flour inequality. In real-world constraints, include equalities if 'at least' or 'no more than' allows it.
A phone battery starts at 100% and loses 12% of its remaining charge each hour. Which expression gives the battery percentage $B$ after $h$ hours?
Answer: $B=100(0.88)^h$ This problem models exponential decay of battery charge over time. Since the battery loses 12% of its remaining charge each hour, it retains 88% (100% - 12% = 88%), giving a decay factor of 0.88. The exponential model B = 100 $(0.88)^h$ correctly represents this, starting at 100% when h = 0. A linear model like B = 100 - 12h would incorrectly subtract 12% of the original charge each hour, eventually going negative. The key insight is that percentage-based changes are multiplicative, not additive. When a quantity changes by a constant percentage, always use an exponential model with base (1 ± rate).
The relationship between temperature in degrees Celsius $C$ and degrees Fahrenheit $F$ is $F=\frac{9}{5}C+32$ . Rewrite this equation to solve for $C$ in terms of $F$ .
Answer: $C=\frac{5}{9}(F-32)$ We need to solve F = (9/5)C + 32 for C in terms of F. First, subtract 32 from both sides: F - 32 = (9/5)C. Then multiply both sides by 5/9 to isolate C: C = (5/9)(F - 32). This is the standard formula for converting Fahrenheit to Celsius. Common errors include forgetting to subtract 32 first or using 9/5 instead of 5/9 when inverting. When solving equations with fractions, remember that multiplying by the reciprocal undoes the original multiplication.
$2x - y = 0$ $x + y = 9$ Which ordered pair $(x, y)$ satisfies both equations?
Answer: $(3,\,6)$ The first equation gives $y = 2x$ . Substituting into the second equation gives $x + 2x = 9$ , so $3x = 9$ and $x = 3$ . Then $y = 2(3) = 6$ , so the solution is $(3, 6)$ . Check: $2(3) - 6 = 0$ and $3 + 6 = 9$ . Every choice satisfies $x + y = 9$ , so the first equation is what separates them: $(6, 3)$ gives $2(6) - 3 = 9$ , $(9, 0)$ gives $18$ , and $(0, 9)$ gives $-9$ , none of which equal $0$ . Always check a candidate pair in both equations.
A cafeteria tracked the number of sandwiches sold each day for 5 days. The line graph shows: Mon 90, Tue 110, Wed 100, Thu 120, Fri 130. According to the graph, what is the total number of sandwiches sold from Tuesday through Thursday?
Answer: 330 sandwiches The question asks for the total sandwiches sold from Tuesday through Thursday. According to the graph: Tuesday = 110, Wednesday = 100, Thursday = 120. Adding these: 110 + 100 + 120 = 330 sandwiches. A common error would be including Monday or Friday in the calculation, or making an arithmetic error when adding the three values.
Triangle $PQR$ has side lengths $PQ=6$ , $QR=10$ , and $PR=x$ . Which value of $x$ is possible for a triangle with these side lengths?
Answer: 12 The question asks which value of x is possible for side PR in triangle PQR with PQ=6 and QR=10. The triangle inequality theorem states that the sum of any two sides must exceed the third side. Check inequalities: x < 6 + 10 = 16, x > 10 - 6 = 4, and also 6 + 10 > x (same as first). Among choices, 12 satisfies 4 < 12 < 16, while 3 is too small, 4 is marginal (degenerate), and 16 is equal (degenerate). Errors include ignoring one inequality or allowing equalities for non-degenerate triangles. Always test all three inequalities for side lengths.
A cone has radius $6\text{ m}$ and height $9\text{ m}$ . What is the volume of the cone? Use $\pi$ in your answer.
Answer: $108\pi\text{ m}^3$ This problem asks for the volume of a cone with radius 6 m and height 9 m. The formula for the volume of a cone is V = ⅓πr²h. Substituting the values: V = ⅓ × π × 6² × 9 = ⅓ × π × 36 × 9 = ⅓ × 324π = 108π m³. A common mistake is forgetting the ⅓ factor, which would give 324π m³. Remember that a cone's volume is exactly one-third of a cylinder with the same base and height.
A store tracked the number of customers per hour for 9 hours: 12, 15, 15, 16, 16, 18, 20, 20, 50. The manager wants a measure of center that best represents a typical hour, given the unusually large value. Which measure is most appropriate: mean or median? Choose the statement that correctly compares them in the presence of the outlier.
Answer: Median is better because it is less affected by outliers This question asks which measure of center better represents typical customer traffic when there's an outlier (50 customers). The mean is (12+15+15+16+16+18+20+20+50)/9 = 182/9 ≈ 20.2 customers. The median is the 5th value when ordered: 12, 15, 15, 16, 16, 18, 20, 20, 50, so median = 16 customers. The outlier of 50 pulls the mean up significantly (to 20.2) while the median remains at 16, which better represents the typical hour. The median is less affected by outliers because it only depends on the middle position(s), not the actual values. When data contains outliers, the median often provides a better measure of central tendency than the mean.
A right triangle has legs of lengths $5\sqrt{3}$ and $5$ . What is the length of the hypotenuse?
Answer: $10$ This problem asks for the hypotenuse of a right triangle with legs $5\sqrt{3}$ and $5$ . Using the Pythagorean theorem: $c^2 = (5\sqrt{3})^2 + 5^2 = 25(3) + 25 = 75 + 25 = 100, \text{ so } c = \sqrt{100} = 10.$ This can be seen as a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle scaled by 5, since in the ratio $1 : \sqrt{3} : 2$ , we have $5 : 5\sqrt{3} : 10$ . A common mistake is to incorrectly square $5\sqrt{3}$ , getting $25\sqrt{3}$ instead of 75. Remember that $(a\sqrt{b})^2 = a^2 b$ when squaring expressions with radicals.
Which expression is equivalent to $\dfrac{6x^2-15x}{3x}$ for $x\ne 0$ ?
Answer: $2x-5$ To simplify $\frac{6x^2-15x}{3x}$ for $x \neq 0$ , first factor the numerator. The GCF of $6x^2$ and $-15x$ is $3x$ , so: $6x^2 - 15x = 3x(2x - 5)$ . Now the fraction becomes $\frac{3x(2x-5)}{3x} = 2x - 5$ after canceling the common factor $3x$ . A common mistake is to cancel terms instead of factors, like trying to cancel just the $x$ from $6x^2$ and $3x$ . Always factor completely before canceling.
A business uses the profit model $P=R-C$ , where revenue is $R=120x$ and cost is $C=45x+300$ . If profit is $1{,}050$ , the equation is $120x-(45x+300)=1050$ . What is the value of $x$ ?
Answer: $18$ This problem gives us 120x - (45x + 300) = 1050 to solve for x. First, we distribute the negative sign: 120x - 45x - 300 = 1050. Combining like terms: 75x - 300 = 1050. Adding 300 to both sides: 75x = 1350. Dividing by 75: x = 1350/75 = 18. A common error is forgetting to distribute the negative sign to all terms in the parentheses. When working with profit equations, remember that profit equals revenue minus all costs.
A phone battery is at 100% and loses 12 percentage points every hour while streaming video. Let $B(t)$ be the battery percent after $t$ hours. Which equation models $B(t)$ , and how long until the battery reaches 40%?
Answer: $B=100-12t$ ; 5 hr This problem describes a battery losing 12 percentage points per hour, which is constant subtraction indicating linear decay. The model is B(t) = 100 - 12t, starting at 100% and decreasing by 12 each hour. To find when B = 40: 40 = 100 - 12t, so 12t = 60, giving t = 5 hours. The key distinction is "percentage points" (additive) versus "percent" (multiplicative) - losing 12 percentage points means subtracting 12, not multiplying by 0.88. Common errors include modeling this as exponential decay or solving the equation incorrectly.
Solve the system: $\begin{cases} 7x+3y=1\\ 14x+6y=5 \end{cases}$ How many solutions does the system have?
Answer: $0$ We need to determine how many solutions the system $7x + 3y = 1$ and $14x + 6y = 5$ has. Looking at the coefficients, I notice that $14x + 6y$ is exactly $2(7x + 3y)$ . If we multiply the first equation by 2, we get $14x + 6y = 2$ . But the second equation says $14x + 6y = 5$ . This means we have $14x + 6y = 2$ and $14x + 6y = 5$ , which is impossible - the same expression cannot equal both 2 and 5. Therefore, this system has no solutions (the lines are parallel). The warning about dividing by 2 and assuming the equations match is exactly the trap here - the coefficients are proportional but the constants are not. When checking for parallel or identical lines, always verify both the coefficients AND the constants.
A gym charges a one-time sign-up fee plus a monthly membership fee. After 2 months, the total cost is $70, and after 6 months, the total cost is $150. Assuming the relationship is linear, which equation gives the total cost $C$ (in dollars) after $m$ months?
Answer: $C=20m+30$ The question asks for the linear equation modeling the total cost C in dollars after m months at a gym, given a one-time sign-up fee and monthly fee, with costs of $70 after 2 months and $150 after 6 months. To find the equation, recognize that the sign-up fee is the y-intercept b and the monthly fee is the slope k, so C = k m + b. Set up the system: 2k + b = 70 and 6k + b = 150. Subtract the first equation from the second to get 4k = 80, so k = 20; substitute into the first to find b = 70 - 40 = 30, yielding C = 20m + 30. A key computational error might involve miscalculating the difference in months or costs, such as using 4 months for $80 difference but dividing incorrectly. Another error could be switching the slope and intercept values. A useful test-taking strategy is to plug in the given months into the choices to verify which matches both costs.
Which expression is equivalent to $4x^2-12x+9$ ?
Answer: $(2x-3)^2$ To factor $4x^2-12x+9$ , check if it's a perfect square trinomial. For $(ax+b)^2 = a^2x^2 + 2abx + b^2$ , we need: first term $4x^2 = (2x)^2$ , last term $9 = 3^2$ . The middle term should be $2(2x)(3) = 12x$ , but we have $-12x$ . This matches $(2x-3)^2 = 4x^2 - 12x + 9$ . Students often try regular factoring methods first, but recognizing perfect square patterns saves time and reduces errors.
Which expression is equivalent to $x^2+6x+1$ written in vertex form $a(x-h)^2+k$ ?
Answer: $(x+3)^2-8$ To write x² + 6x + 1 in vertex form, we need to complete the square. First, take half of the coefficient of x and square it: (6/2)² = 9. Add and subtract 9: x² + 6x + 9 - 9 + 1 = (x + 3)² - 8. The vertex form is (x + 3)² - 8, which can also be written as (x - (-3))² + (-8). Common errors include forgetting to subtract the added term (getting (x + 3)² + 10) or making sign errors with the h-value. When completing the square, always remember to balance what you add by subtracting the same amount.
A sector of a circle has radius $5$ in and area $\frac{25\pi}{4}$ in $^2$ . What is the measure of the central angle of the sector, in degrees?
Answer: $90^\circ$ The question asks for the central angle of a sector with radius 5 in and area 25π/4 in². The sector area formula is (θ/360) × πr²; solve for θ. With r = 5, full area π × 25 = 25π, so (θ/360) × 25π = 25π/4, θ/360 = 1/4, θ = 90°. The fraction simplifies directly. A common error is using arc length or forgetting to multiply by 360, leading to 45° or 180°. Another is confusing r with diameter, doubling area. Emphasize isolating θ correctly and using radius in r². Rearrange the formula before substituting to avoid errors.
A car's stopping distance model is $y=x^2-2x$ and a safety guideline is $y=4x-8$ , where $x$ is speed (in tens of mph) and $y$ is distance (in feet). At what points do the graphs intersect? $\begin{cases}y=x^2-2x\\y=4x-8\end{cases}$
Answer: $(2,0)$ and $(4,8)$ We need to find where the car's stopping distance parabola intersects the safety guideline line. This system has a quadratic equation $y = x^2 - 2x$ and a linear equation $y = 4x - 8$ , solvable by substitution. Setting them equal: $x^2 - 2x = 4x - 8$ , which simplifies to $x^2 - 6x + 8 = 0$ . Factoring gives $(x - 2)(x - 4) = 0$ , so $x = 2$ or $x = 4$ . When $x = 2$ , $y = 4(2) - 8 = 0$ ; when $x = 4$ , $y = 4(4) - 8 = 8$ , giving us points $(2, 0)$ and $(4, 8)$ . A common error is mixing up the x and y coordinates or only finding one solution when there are two. Always verify your solutions by substituting back into both original equations.
A line is graphed on a coordinate plane. Moving from the point $(1,3)$ to $(5,1)$ on the line, what is the slope?
Answer: $-\frac{1}{2}$ The question asks for the slope of a line from (1,3) to (5,1). Use m=(1-3)/(5-1)=-2/4=-1/2. This negative slope indicates decrease. The calculation shows a drop of 2 over run of 4. A common error is sign reversal, getting positive 1/2. Another mistake might be dividing incorrectly like -2/2=-1. For slope from points, always subtract y's over x's consistently to understand direction.
A phone plan is modeled by $35+0.08m=50+0.05m$ , where $m$ is the number of text messages sent in a month. What is the value of $m$ when the two plans cost the same?
Answer: $500$ This problem asks when two phone plans cost the same by solving 35 + 0.08m = 50 + 0.05m. First, we collect like terms by subtracting 0.05m from both sides: 35 + 0.08m - 0.05m = 50, which gives us 35 + 0.03m = 50. Subtracting 35 from both sides: 0.03m = 15. Dividing by 0.03: m = 15/0.03 = 500. The key error to avoid is mishandling the decimal coefficients when combining like terms. When comparing plans, the intersection point represents where both options have equal value.
A circle has a radius of 4 cm. What is the area of the circle?
Answer: 16π cm² This question asks for the area of a circle in square centimeters given its radius of 4 cm. The formula for the area of a circle is A = πr². Substituting the given radius: A = π(4)² = π(16) = 16π cm².
A music streaming service charges a monthly fee of $6$ plus $0.50$ for each song downloaded. Let $x$ be the number of songs downloaded in a month and $y$ be the total monthly cost in dollars. Which equation represents this relationship?
Answer: $y=0.5x+6$ This question asks us to write an equation for a linear cost function with a fixed monthly fee and a per-song charge. The total cost y equals the fixed monthly fee ( $6$ ) plus the variable cost ( $0.50$ per song times x songs), giving us $y = 6 + 0.50x$ , which can be written as $y = 0.5x + 6$ . A common error is reversing the coefficients, putting the per-song cost as the constant term. Remember that in $y = mx + b$ form, m is the rate of change (cost per song) and b is the y-intercept (fixed fee).
On a coordinate plane, the points $(0,40)$ , $(1,60)$ , and $(2,90)$ represent revenue $R$ (in dollars) after $t$ hours. Which model is more appropriate, linear or exponential, and why?
Answer: Exponential; ratios are constant This problem gives three points to analyze: (0, 40), (1, 60), and (2, 90). Checking differences: 60 - 40 = 20 and 90 - 60 = 30, which are not constant. Checking ratios: 60/40 = 1.5 and 90/60 = 1.5, which are constant. Since the ratios are constant at 1.5, this represents exponential growth where revenue increases by 50% each hour. The model would be R = 40 $(1.5)^t$ . The key insight is that constant ratios indicate exponential growth, while constant differences indicate linear growth. Always check both to determine the appropriate model type.
A real estate agent compared house size (square feet) and sale price (thousands of dollars) for 12 houses in one neighborhood. The scatterplot shows a line of best fit. Using the line of best fit, what is the predicted sale price for a $2{,}200$ square-foot house?
Answer: $\$300{,}000$ This question asks for the predicted sale price of a 2,200 square-foot house using the line of best fit. Looking at the scatterplot where house size is in square feet and price is in thousands of dollars, we find x = 2,200 on the horizontal axis and trace up to the line. At x = 2,200 square feet, the line appears to intersect at approximately y = 300 (thousands of dollars), which equals $300,000. The positive relationship shows larger houses tend to sell for higher prices, as expected. Remember that the y-axis shows thousands of dollars, so a reading of 300 means $300,000, not $300.
A company's quarterly revenue is shown in the line graph (in millions of dollars) for Q1 through Q4. A note from accounting says Q2 revenue included a one-time $1 million refund that reduced the reported value. Based on the graph, what would Q2 revenue have been without the refund? Add the refund back to the Q2 value shown.
Answer: $11 million The question states that Q2 revenue was reduced by a $1 million refund and asks what it would have been without the refund. From the graph, Q2 shows $10 million. Adding back the refund: $10 million + $1 million = $11 million. A common error is subtracting the refund instead of adding it back, or misunderstanding that the graph already shows the reduced value.
A rectangular garden is $2.4$ meters long and $150$ centimeters wide. Using $100\text{ cm}=1\text{ m}$ , what is the area of the garden in square meters?
Answer: 3.6 m $^2$ We need to find the area of a garden that is 2.4 m long and 150 cm wide in square meters. First convert the width: 150 cm × (1 m/100 cm) = 1.5 m. The area is 2.4 m × 1.5 m = 3.6 m². A common error is forgetting to convert centimeters to meters before multiplying, or incorrectly converting 150 cm. When finding area with mixed units, always convert all measurements to the same unit before multiplying.
A store offers a discount modeled by $p=0.75m+6$ , where $m$ is the marked price and $p$ is the price paid. If a customer paid $36$ dollars, what was the marked price $m$ ?
Answer: $40$ This problem asks us to find the marked price m when the paid price is $36 using p = 0.75m + 6. We substitute p = 36: 36 = 0.75m + 6. Subtracting 6 from both sides: 30 = 0.75m. Dividing by 0.75: m = 30/0.75 = 40. A common mistake is to incorrectly divide by the decimal coefficient. When working with discount formulas, remember that the coefficient represents the percentage paid after discount.
A line on the coordinate plane passes through the points $(-2,3)$ and $(4,-9)$ . What is the slope of the line?
Answer: $-2$ To find the slope between points (-2, 3) and (4, -9), we use the slope formula: m = (y₂ - y₁)/(x₂ - x₁). Substituting: m = (-9 - 3)/(4 - (-2)) = -12/6 = -2. The negative slope indicates the line decreases from left to right. Common errors include subtracting coordinates in different orders for numerator and denominator or making sign errors with negative coordinates. When calculating slope, maintain consistent order and carefully handle negative values.
A taxi charges a base fare plus a per-mile rate. A 3-mile ride costs $11.50, and a 9-mile ride costs $23.50. If the cost $C$ is linear in miles $m$ , what is the base fare (in dollars)?
Answer: $5.50$ The question asks for the base fare of a taxi service that charges a fixed amount plus a per-mile rate, given a 3-mile ride costs $11.50 and a 9-mile ride costs $23.50, assuming a linear relationship. To solve, set up equations with base fare b and per-mile rate p: b + 3p = 11.50 and b + 9p = 23.50. Subtract the first from the second: 6p = 12, so p = 2; then b = 11.50 - 6 = 5.50. This identifies the base fare as $5.50. A key error could be forgetting to subtract correctly, leading to wrong p like 3.50 if dividing 12 by 3.4 or similar miscalculation. Another common mistake is solving for p instead of b. When finding intercepts in linear models, isolate the fixed component by eliminating the variable rate.
What are the solutions to the equation $x^2-5x-24=0$ ? Choose the pair of real solutions that makes the equation true.
Answer: $x=-3,\ 8$ This question asks us to find the solutions to the quadratic equation $x^2-5x-24=0$ . To solve by factoring, we need two numbers that multiply to $-24$ (not $+24$ ) and add to $-5$ . These numbers are $-8$ and $+3$ , giving us $(x-8)(x+3)=0$ . Setting each factor to zero: $x-8=0$ gives $x=8$ , and $x+3=0$ gives $x=-3$ . The common error of using factors of $+24$ would lead to incorrect signs. When factoring quadratics with a negative constant term, remember one factor must be positive and one negative.
At what time does the object reach maximum height if its height is modeled by $h(t) = -16t^2 + 32t + 5$ ?
Answer: 1 sec This question asks when an object reaches maximum height, which occurs at the vertex of the parabola. For a quadratic in the form $h(t) = at^2 + bt + c$ , the time at maximum height is $t = -\frac{b}{2a}$ . With $h(t) = -16t^2 + 32t + 5$ , we have $a = -16$ and $b = 32$ , so $t = -\frac{32}{2(-16)} = -\frac{32}{-32} = 1$ second. A common error is forgetting the negative sign in the formula or mixing up the values of $a$ and $b$ . For projectile motion problems, always use the vertex formula to find the time of maximum height.
Solve the system. What is the solution $(x, y)$ ? $\begin{cases} 5x + y = 14 \\ 2x - 3y = -1 \end{cases}$
Answer: $(\tfrac{41}{17},\,\tfrac{33}{17})$ We need to solve the system $5x + y = 14$ and $2x - 3y = -1$ . I'll use elimination to eliminate y by multiplying the first equation by 3: $15x + 3y = 42$ . Adding this to the second equation: $15x + 3y + 2x - 3y = 42 + (-1)$ , which gives $17x = 41$ , so $x = \frac{41}{17}$ . Substituting back into the first equation: $5(\frac{41}{17}) + y = 14$ , so $\frac{205}{17} + y = 14$ . Converting 14 to seventeenths: $\frac{238}{17}$ , we get $y = \frac{238}{17} - \frac{205}{17} = \frac{33}{17}$ . The common error mentioned would be adding without proper alignment, like adding $5x + y + 2x - 3y$ incorrectly. When solving systems, always align like terms vertically before adding or subtracting equations.
The expression $4x^2 + 12x + 9$ can be rewritten as $(2x + b)^2$ . What is the value of $b$ ?
Answer: 3 This question asks us to find the value of $b$ by recognizing a perfect square trinomial pattern. We need $(2x + b)^2$ to equal $4x^2 + 12x + 9$ . Expanding $(2x + b)^2$ gives us $4x^2 + 4bx + b^2$ . Comparing this to $4x^2 + 12x + 9$ , we need $4b = 12$ and $b^2 = 9$ . Both equations give us $b = 3$ .
A phone battery has 100% charge at $t=0$ and loses 12 percentage points per hour. Let $B(t)$ be the battery percentage after $t$ hours. Which equation models $B(t)$ ?
Answer: $B(t)=100-12t$ A battery starts at 100% and loses 12 percentage points per hour. "Loses 12 percentage points" indicates subtraction (linear decay), not multiplication (exponential decay). The model is B(t) = 100 - 12t, where we subtract 12t from the initial 100%. We can verify: at t=1, B(1) = 100-12(1) = 88%; at t=2, B(2) = 100-12(2) = 76%. The key distinction is between losing a constant amount (linear) versus losing a constant percentage of the remaining amount (exponential).
In the coordinate plane, points $A(-3,0)$ , $B(1,4)$ , and $C(5,0)$ form triangle $ABC$ . If $AB=BC$ , what is the measure of $\angle B$ ?
Answer: $90^\circ$ This problem involves finding an angle in an isosceles triangle given coordinates. With A(-3,0), B(1,4), and C(5,0), and AB = BC, we need angle B. First, let's verify AB = BC using the distance formula: AB = √[(1-(-3))² + (4-0)²] = √[16 + 16] = √32 = 4√2, and BC = √[(5-1)² + (0-4)²] = √[16 + 16] = √32 = 4√2 ✓. Notice that A and C both lie on the x-axis (y = 0), and B is directly above the midpoint of AC. The midpoint of AC is ((−3+5)/2, 0) = (1, 0), and B is at (1, 4), confirming B is directly above. This creates an isosceles triangle where the altitude from B to AC is perpendicular to AC, making angle B = 90°. A quick way to recognize this: when an isosceles triangle has its vertex directly above the midpoint of the base, the vertex angle is 90°.
Two students tracked their daily step counts for 5 days. Student A: 4,000; 6,000; 6,500; 7,000; 26,000. Student B: 5,500; 6,000; 6,500; 7,000; 7,500. Each student reports a mean. Which statement is true about how the outlier affects the means and medians?
Answer: A's mean is much larger than A's median; B's mean and median are close. The question asks which statement is true about how the outlier affects the means and medians of two students' step counts. For Student A, calculate the mean as (4,000 + 6,000 + 6,500 + 7,000 + 26,000)/5 = 13,700 and median as the 3rd value (6,500); for Student B, mean = (5,500 + 6,000 + 6,500 + 7,000 + 7,500)/5 = 6,500 and median = 6,500. The outlier 26,000 inflates A's mean far above its median, while B's mean and median are equal and close. This illustrates the mean's sensitivity to extremes compared to the median. A common error is assuming outliers equally affect both measures, but the median remains stable. When comparing datasets, compute both center measures to see outlier impacts clearly.
A right triangle has legs $2x$ and $3x$ and hypotenuse $6a3$ . What is the value of $x$ ?
Answer: $6$ We have a right triangle with legs 2x and 3x and hypotenuse 6√13. Applying the Pythagorean theorem: (2x)² + (3x)² = (6√13)², which gives us 4x² + 9x² = 36(13) = 468. Combining like terms: 13x² = 468, so x² = 36, and therefore x = 6. Students often make errors when squaring expressions with radicals or forget to square the coefficient along with the variable. To verify your answer, substitute back: legs are 12 and 18, and 12² + 18² = 144 + 324 = 468 = (6√13)².
A map uses a scale of $1:24{,}000$ , meaning $1$ unit on the map equals $24{,}000$ of the same units in reality. Two trailheads are $3.6\text{ in}$ apart on the map. Approximately how many miles apart are they in reality? (Use $1\text{ mile}=63{,}360\text{ in}$ .)
Answer: 1.36 mi This question requires finding the actual distance in miles between two trailheads that are 3.6 in apart on a map with a scale of 1:24,000. Set up the proportion: actual distance (in) = 3.6 in × 24,000. Calculate 3.6 × 24,000 = 86,400 in, then convert to miles: 86,400 / 63,360 ≈ 1.36 mi. This scales the map distance and converts units carefully. A common error is forgetting to convert inches to miles, leaving the answer in inches. Emphasize unit conversions in scale problems to match the requested units. For strategy, approximate first (3.6 × 24,000 / 63,360 ≈ 3.6 × 0.379 ≈ 1.36) to check reasonableness.
A right triangle has acute angles $\theta$ and $90^\circ-\theta$ . If $\tan(\theta)=\frac{2}{7}$ , what is $\tan(90^\circ-\theta)$ ?
Answer: $\frac{7}{2}$ This problem uses the complementary angle relationship for tangent. If $tan(\theta) = \frac{2}{7}$ , then $tan(90^\circ - \theta)$ equals the reciprocal: $tan(90^\circ - \theta) = \frac{7}{2}$ . This follows from the cofunction identity: $tan(90^\circ - \theta) = cot(\theta) = \frac{1}{tan(\theta)}$ . Since $tan(\theta) = \frac{2}{7}$ , we have $tan(90^\circ - \theta) = \frac{1}{\frac{2}{7}} = \frac{7}{2}$ . A common mistake is thinking $tan(90^\circ - \theta) = -tan(\theta)$ or forgetting the reciprocal relationship. Remember that complementary angles in a right triangle swap the roles of opposite and adjacent sides.
A polynomial is given by $r(x)=(x^2-9)(x^2+4x+4)$ . How many distinct real zeros does $r(x)$ have?
Answer: 3 The question asks for the number of distinct real zeros of $r(x) = (x^2 - 9)(x^2 + 4x + 4)$ . Factor: $x^2 - 9 = (x - 3)(x + 3)$ , $x^2 + 4x + 4 = (x + 2)^2$ . Zeros are $x = 3$ , $-3$ , and $-2$ (repeated). Distinct real zeros: 3. This factorization reveals roots and multiplicities, linking algebraic form to unique solutions despite repetition. A common error counts the repeated zero as two distinct or includes complex ones. Some miscount by not fully factoring. Strategically, solve each factor separately and use a set to count uniques.

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