SAT Practice Test 5 - SAT Math

A weather station recorded the average high temperature each day for 6 days. The line graph shows the temperatures for Mon through Sat. According to the graph, by how many degrees Fahrenheit did the temperature change from Wednesday to Friday?
Answer: Decreased by $10^\circ\text{F}$ The question asks by how many degrees Fahrenheit the average high temperature changed from Wednesday to Friday based on the line graph. On the graph, the point for Wednesday is at 60°F, and for Friday at 50°F, with the y-axis in 5°F increments, so use the exact plotted points rather than estimating between them. The change is 50 - 60 = -10°F, which means a decrease of 10°F. Be careful not to confuse the direction of change; a drop is a decrease, not an increase. A key error could be misreading the scale and thinking the difference is 5°F instead, or using a trend line instead of actual points as instructed.
A box contains 5 red balls and 3 blue balls. If one ball is drawn at random, what is the probability that it is a red ball?
Answer: $\frac{5}{8}$ This question asks for the probability of drawing a red ball from a box containing red and blue balls. The favorable outcomes are the 5 red balls, and the total possible outcomes are all 8 balls (5 red + 3 blue). Using the probability formula $P(\text{red}) = \frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{5}{8}$ .
In a right triangle, one angle measures $30^\text{o}$ . What is the measure of the other non-right angle?
Answer: $60^\text{o}$ This question involves finding the third angle in a right triangle when one acute angle is given. In any triangle, the sum of all angles equals 180°, and in a right triangle, one angle is exactly 90°. Given that one angle is 30° and another is 90°, the third angle measures 180° - 90° - 30° = 60°. This creates a special 30-60-90 triangle, which is commonly tested. Students sometimes forget that 'right triangle' means one angle is exactly 90°, or they might add angles incorrectly. Remember that right triangles always have one 90° angle plus two acute angles that sum to 90°.
For the function $f(x) = 4x - 7$ , what is the value of $f(3)$ ?
Answer: 5 This question asks us to evaluate a linear function at a specific input value. For f(x) = 4x - 7, we need to find f(3) by substituting x = 3 into the function. This gives us f(3) = 4(3) - 7 = 12 - 7 = 5. Function evaluation requires careful substitution and following the order of operations (multiplication before subtraction). A typical computational error would be incorrectly calculating 4 × 3 or making a mistake with the subtraction 12 - 7. When evaluating functions, substitute the input value carefully and perform arithmetic operations step by step.
A phone plan cost $C$ is modeled by $C=15+0.08m$ , where $m$ is the number of minutes. Solve for $m$ in terms of $C$ . Which expression gives $m$ ?
Answer: $m=\frac{C-15}{0.08}$ We need to solve $C=15+0.08m$ for $m$ in terms of $C$ . First, subtract 15 from both sides: $C - 15 = 0.08m$ . To isolate $m$ , divide both sides by 0.08: $m = \frac{C - 15}{0.08}$ . This gives us the number of minutes in terms of the total cost. A common mistake is incorrectly handling the order of operations, such as dividing only $C$ by 0.08 instead of the entire expression $(C - 15)$ . When solving for a variable, always isolate it step by step.
A model airplane is built using a scale of 1:50. If the wingspan of the model is 0.5 meters, what is the actual wingspan of the airplane?
Answer: 25 meters This question involves finding the actual wingspan when given a scale model measurement and the scale ratio. The scale 1:50 means 1 unit on the model represents 50 units in reality. We can set up the proportion: model/actual = 1/50, so 0.5/x = 1/50. Cross-multiplying gives us x = 0.5 × 50 = 25 meters.
A population of deer is shown in the graph of $P$ versus $t$ (years). The curve rises slowly at first and then more rapidly. Which model is more appropriate, and which feature of the graph supports your choice?
Answer: Exponential; increasing rate of change This question describes a graph that rises slowly at first and then more rapidly, which is characteristic of exponential growth. Exponential functions show an increasing rate of change - the curve gets steeper as time progresses because each increase is a percentage of the current value. Linear functions maintain a constant rate of change and would appear as a straight line. The key feature supporting exponential growth is the accelerating rate of increase. A common error is choosing linear because of confusion about what "constant ratio" means. For visual identification: if the curve bends upward (concave up), it's likely exponential growth.
Solve the system $\begin{cases}y=2x-5\\y=-x+4\end{cases}$ and determine the $x$ -coordinate of the solution. What is the $x$ -coordinate?
Answer: $3$ We need to find where the lines $y = 2x - 5$ and $y = -x + 4$ intersect, specifically the x-coordinate. I'll use substitution since both equations are already solved for y. Setting the expressions equal: $2x - 5 = -x + 4$ . Adding x to both sides: $3x - 5 = 4$ . Adding 5 to both sides: $3x = 9$ , so $x = 3$ . The warning about the common mistake is important - if you incorrectly wrote $2x - 5 = -x - 4$ , you'd get $3x = 1$ , giving $x = 1/3$ . Always copy equations carefully when setting them equal.
A fitness app tracked the number of steps a user walked each day for 6 days. The bar graph shows steps (in thousands) for Day 1 through Day 6. What is the total number of steps (in thousands) for Day 2, Day 4, and Day 6 combined?
Answer: 20 thousand The question asks for the total steps (in thousands) for Days 2, 4, and 6 combined. Reading from the bar graph: Day 2 shows 5 thousand, Day 4 shows 7 thousand, and Day 6 shows 8 thousand steps. The total is 5 + 7 + 8 = 20 thousand steps. Students often misread the scale or accidentally include adjacent days in their calculation.
What is the volume of a cylinder with a height of 10 cm and a diameter of 6 cm?
Answer: 90π cm³ This question asks for the volume of a cylinder with height 10 cm and diameter 6 cm. The formula for cylinder volume is V = πr²h. Since the diameter is 6 cm, the radius is 3 cm. Substituting: V = π(3)²(10) = π(9)(10) = 90π cm³.
Two data sets have the same mean of 50. Data set A is: 49, 50, 50, 51. Data set B is: 20, 50, 50, 80. Which statement best compares the standard deviations of the two sets? Consider standard deviation as a measure of spread around the mean.
Answer: Set B has greater standard deviation. This question compares standard deviations of two datasets with the same mean (50). Set A: 49, 50, 50, 51 has values very close to the mean. Set B: 20, 50, 50, 80 has values much farther from the mean. Standard deviation measures spread around the mean - the farther values are from the mean, the larger the standard deviation. Set B clearly has greater spread (values differ from mean by up to 30) compared to Set A (values differ by at most 1). Therefore, Set B has the greater standard deviation. When comparing standard deviations visually, look at how far values spread from the center.
Two parallel lines are cut by a transversal. An alternate interior angle measures $121^\circ$ . Angle $x$ is the interior angle on the same intersection that forms a linear pair with that $121^\circ$ angle. What is the measure of $x$ ?
Answer: $59^\circ$ This question asks for an angle that forms a linear pair with an alternate interior angle. The alternate interior angle is 121°, and angle x forms a linear pair with this angle at the same intersection. Linear pairs are supplementary, so x + 121° = 180°, giving us x = 180° - 121° = 59°. A common mistake is confusing the relationship and thinking x should equal 121° because of the parallel lines. Remember that linear pairs always sum to 180° regardless of parallel lines.
Which expression is equivalent to $(2x+5)^2$ ? Expand using the perfect square identity $(a+b)^2=a^2+2ab+b^2$ .
Answer: $4x^2+20x+25$ To expand $(2x+5)^2$ , we use the perfect square identity $(a+b)^2 = a^2 + 2ab + b^2$ where $a = 2x$ and $b = 5$ . First term: $(2x)^2 = 4x^2$ ; Middle term: $2(2x)(5) = 20x$ ; Last term: $5^2 = 25$ . Therefore, $(2x+5)^2 = 4x^2 + 20x + 25$ . A common error is calculating the middle term as just $2x \cdot 5 = 10x$ instead of $2 \cdot 2x \cdot 5 = 20x$ , forgetting the factor of 2 in the formula. Remember that the middle term in a perfect square expansion is always twice the product of the two terms.
A ladder is leaning against a wall, forming a right triangle with the ground. If the base of the ladder is 3 meters from the wall and the ladder is 5 meters long, what is the height at which the ladder touches the wall?
Answer: 4 meters This problem asks for the height where a ladder touches a wall, given the ladder length and distance from the wall. We use the Pythagorean theorem since the ladder, wall, and ground form a right triangle. With the ladder as hypotenuse (5 meters) and the base distance as one leg (3 meters), we find the height: 3² + h² = 5², so 9 + h² = 25, which gives h² = 16 and h = 4 meters.
What can be concluded from the study?
Answer: Those who meditate longer tend to report better sleep quality. This question asks what can be concluded from survey results showing a positive correlation between meditation duration and sleep quality scores. The data indicates that participants who meditate for longer periods tend to report better sleep quality, establishing an association within the sample. Choice B correctly identifies this trend using appropriate language ("tend to") that reflects the correlational nature of the findings. Choice A overgeneralizes to "all participants," choice C incorrectly claims direct causation, and choice D contradicts the positive correlation findings. When analyzing survey data, distinguish between observed patterns in the sample versus claims about causation or universal effects.
A plant is 10 cm tall and grows by 8% each week. Which expression gives its height after $w$ weeks?
Answer: $10(1.08)^w$ Gaining 8% per week is repeated multiplication, not repeated addition: each week the plant keeps 100% of its height and adds 8%, a factor of 1.08. Starting at 10 cm, its height after w weeks is 10 $(1.08)^w$ , and checking w = 1 gives 10.8 cm, which is indeed 8% above 10 cm. The expression 10 + 0.08w models linear growth of only 0.08 cm per week, adding a fixed sliver instead of 8% of the current height. The expression 10 $(8)^w$ multiplies by 8 every week, a 700% increase that would put the plant at 80 cm after one week. The expression 10 $(0.92)^w$ uses the factor for 8% decay, so it shrinks the plant to 9.2 cm after a week rather than growing it.
A coordinate plane shows the graphs of $2x+y=8$ and $x-y=1$ . What is the value of $x-y$ at their intersection point?
Answer: $1$ We need to find x - y at the intersection of 2x + y = 8 and x - y = 1. The question cleverly asks for x - y, which is already given as 1 in the second equation! Since the second equation states that x - y = 1 for all points on that line, and we want the value at the intersection point (which must satisfy both equations), the answer is simply 1. To verify this makes sense, I'll solve for the intersection: from x - y = 1, we get x = y + 1. Substituting into 2x + y = 8: 2(y + 1) + y = 8, so 3y + 2 = 8, thus y = 2 and x = 3. Indeed, x - y = 3 - 2 = 1 ✓.
On the coordinate plane, a line passes through the points $(-4,5)$ and $(2,-1)$ . Which equation gives the line in slope-intercept form?
Answer: $y = -x + 1$ We need to find the equation of the line through (-4, 5) and (2, -1) in slope-intercept form (y = mx + b). First, calculate the slope: m = (-1 - 5)/(2 - (-4)) = -6/6 = -1. Now use point-slope form with either point; using (2, -1): y - (-1) = -1(x - 2), which gives y + 1 = -x + 2, so y = -x + 1. We can verify by checking both points: for (-4, 5): y = -(-4) + 1 = 4 + 1 = 5 ✓; for (2, -1): y = -(2) + 1 = -2 + 1 = -1 ✓. A common error is sign confusion when subtracting negative coordinates. Always double-check your final equation with both given points.
If $2(3x+k)-5x$ is equivalent to $x+8$ for all values of $x$ , what is the value of $k$ ?
Answer: $4$ We need to find $k$ such that $2(3x+k)-5x$ equals $x+8$ for all values of $x$ . First, expand the left side: $2(3x+k)-5x = 6x+2k-5x = x+2k$ . For this to equal $x+8$ , we need the coefficients of $x$ and the constants to match separately. The $x$ coefficients already match (both are 1), and the constants give us $2k = 8$ , so $k = 4$ . A common error is confusing which terms to match or solving $6x+2k-5x = 8$ incorrectly by mixing the $x$ terms with constants.
Which equation is equivalent to $y=x^2+6x+5$ written in vertex form?
Answer: $y=(x+3)^2-4$ We need to convert $y=x^2+6x+5$ to vertex form by completing the square. Starting with $y=x^2+6x+5$ , we take half of the coefficient of $x$ (which is 3) and square it (getting 9). We rewrite as $y=(x^2+6x+9)-9+5=(x+3)^2-4$ . The vertex form is $y=(x+3)^2-4$ , showing the vertex at $(-3,-4)$ . A common error is forgetting to subtract the 9 we added when completing the square. Always verify by expanding your vertex form to ensure it matches the original equation.
In the coordinate plane, a circle has center $(2,-1)$ and passes through the point $(8,3)$ . What is the equation of the circle?
Answer: $(x-2)^2+(y+1)^2=52$ The question asks for the equation of a circle with center (2, -1) passing through (8, 3). The standard form is (x-h)² + (y-k)² = r², where (h,k) is the center; find r using the distance from center to the point. Calculate r = √[(8-2)² + (3 - (-1))²] = √[6² + 4²] = √[36 + 16] = √52. Thus, the equation is (x-2)² + (y+1)² = 52. A common error is switching signs for the center, like using (y-1) instead of (y+1), or squaring incorrectly to get 20 instead of 52. Emphasize radius as the distance, not diameter. Plug the point back into the equation to verify it satisfies it.
$\sqrt[3]{p^{2}}=t^{\frac{4}{9}}$ In the given equation, $p$ and $t$ are constants greater than $1$ . If $t=p^{3n}$ , where $n$ is a constant, what is the value of $n$ ?
Answer: $\dfrac{1}{2}$ Rewrite the left side as $p^{2/3}$ . Substituting $t=p^{3n}$ into the right side gives $t^{4/9}=(p^{3n})^{4/9}=p^{4n/3}$ . Since $p>1$ , the exponents must be equal: $2/3=4n/3$ , so $2=4n$ and $n=1/2$ . A common error is cross-multiplying incorrectly or forgetting to multiply the exponents when raising a power to a power.
A runner tracked distance run each week for 4 weeks. The line graph shows Week 1 through Week 4 distances in miles. According to the graph, what was the average (mean) distance per week over the 4 weeks?
Answer: 4.0 miles The question asks for the average distance run per week over 4 weeks, based on the line graph. The plotted points show Week 1 at 3.5 miles, Week 2 at 4.0, Week 3 at 4.5, and Week 4 at 4.0, so sum these values: 3.5 + 4.0 + 4.5 + 4.0 = 16 miles. Divide by 4 to find the mean: 16 ÷ 4 = 4.0 miles. Use the exact points, not a trend. A key error is calculating the median instead of the mean or misreading one point, like confusing 4.5 with 5.0 due to scale issues.
A line has slope $-\dfrac{3}{4}$ and passes through the point $ (8,-1) $. Which equation is the line in point-slope form?
Answer: $y+1=-\dfrac{3}{4}(x-8)$ The question asks for the point-slope form of the line with slope -3/4 passing through (8, -1). The point-slope form is $ y - y1 = m(x - x1) $, so $ y - (-1) = -3/4 (x - 8) $, or $ y + 1 = -3/4 (x - 8) $. This form is useful for writing equations when given a point and slope. It emphasizes the linear structure around a specific point. A common error is using the wrong sign for the y-coordinate, like y - 1 instead of y + 1. Another mistake might be inverting the slope fraction. A strategy is to recall that point-slope form directly incorporates the given point and slope without needing to solve for intercept first.
A gym charges a one-time sign-up fee plus a monthly fee. After 4 months the total cost is modeled by $25+4x=73$ , where $x$ is the monthly fee in dollars. What is the value of $x$ ?
Answer: $12$ We need to find the monthly fee $x$ from the equation $25 + 4x = 73$ . To isolate $x$ , first subtract 25 from both sides: $4x = 73 - 25 = 48$ . Then divide both sides by 4: $x = 48 ÷ 4 = 12$ . The monthly fee is $12. A common error is dividing 73 by 4 instead of first subtracting the sign-up fee. When solving word problems, always identify what each term represents before manipulating the equation.
Two data sets each have 5 values. Data set A: 4, 6, 6, 8, 26. Data set B: 4, 6, 6, 8, 10. Which statement is true about how the outlier in data set A affects the mean and median compared to data set B?
Answer: A has larger mean but same median This question compares how an outlier (26 in set A vs 10 in set B) affects mean and median. For set A: mean = (4+6+6+8+26)÷5 = 50÷5 = 10; median = 6 (middle of 4,6,6,8,26). For set B: mean = (4+6+6+8+10)÷5 = 34÷5 = 6.8; median = 6 (middle of 4,6,6,8,10). The median is the same (6) for both sets because the middle value doesn't change. However, the mean of A (10) is larger than the mean of B (6.8) because the outlier 26 pulls the mean up significantly more than 10 does. This demonstrates that outliers affect the mean but may not affect the median at all. When comparing datasets with outliers, always calculate both measures to see the full picture.
A water tank is being filled at a constant rate. After 2 hours, the tank contains 150 gallons, and after 5 hours, it contains 300 gallons. What is the rate at which the tank is being filled?
Answer: 50 gallons per hour This problem asks for the constant rate of filling, which is the slope of the linear relationship between time and water volume. Using two points from the given information: (2, 150) and (5, 300), we apply the slope formula: rate = (300 - 150)/(5 - 2) = 150/3 = 50 gallons per hour. This rate represents how much water is added per hour, which is the change in volume divided by the change in time. A common error would be using the total volume instead of the change in volume, or mixing up the time values. In rate problems, always focus on the change in quantity over the change in time to find the constant rate.
A store discounts an item so that the sale price is 85% of the original price. If the sale price is $51, what was the original price?
Answer: $60.00 This problem states that the sale price ($51) is 85% of the original price, and we need to find the original price. We can write this as 0.85 × original price = $51, so original price = $51 ÷ 0.85. Calculating: $51 ÷ 0.85 = $60. A common error is to calculate 85% of $51 instead of dividing, or to subtract 15% from $51. When a sale price is given as a percentage of the original, divide the sale price by the decimal form of the percentage to find the original price.
A sphere has a radius of 5 cm. What is the volume of the sphere?
Answer: 500π/3 cm³ This problem asks for the volume of a sphere with radius 5 cm, requiring the result in cubic centimeters. The formula for the volume of a sphere is V = (4/3)πr³, where r is the radius. Substituting the given radius: V = (4/3)π(5)³ = (4/3)π(125) = (4 × 125π)/3 = 500π/3 cm³. A common error is using the surface area formula (4πr²) or forgetting the (4/3) coefficient. Remember that sphere volume involves r³ with the specific coefficient (4/3)π.
Which of the following expressions is equivalent to $\dfrac{x^2-16}{x-4}$ for all $x \neq 4$ ?
Answer: $x+4$ Factor the numerator as a difference of squares: $x^2-16=(x-4)(x+4)$ . Dividing by $x-4$ (valid since $x\neq 4$ ) leaves $x+4$ . Choice A mistakenly keeps the factor being canceled instead of the remaining one. Choices C and D come from squaring one of the factors instead of simplifying the quotient.
A formula for converting units is rearranged into $\frac{5}{9}(x-32)=10$ . What is the value of $x$ ?
Answer: $50$ We need to solve $\frac{5}{9}(x - 32) = 10$ by isolating x. First, multiply both sides by $\frac{9}{5}$ to clear the fraction: $x - 32 = 10 \times \frac{9}{5} = \frac{90}{5} = 18$ . Then add 32 to both sides to get $x = 18 + 32 = 50$ . A common mistake is to multiply by $\frac{5}{9}$ instead of its reciprocal $\frac{9}{5}$ , which would give $x - 32 = 10 \times \frac{5}{9} = \frac{50}{9} \approx 5.56$ , leading to $x \approx 37.56$ . When a fraction multiplies a parenthetical expression, divide both sides by that fraction (multiply by its reciprocal) to isolate the parentheses.
A streaming service tracks the number of subscribers $S$ (in thousands) after $t$ months. The data follow a pattern where each month the total increases by the same amount. If $S=18$ when $t=0$ and $S=30$ when $t=4$ , which equation best models the relationship between $S$ and $t$ , and what does its key parameter represent?
Answer: $S=3t+18$ ; +3 thousand per month This question asks for a model of subscriber growth where the total increases by the same amount each month, indicating a linear relationship. Since the increase is constant each month, we need a linear model of the form S = mt + b, where m is the monthly increase and b is the initial value. From the given data, when t = 0, S = 18, so b = 18. When t = 4, S = 30, giving us 30 = 4m + 18, which solves to m = 3. Therefore, S = 3t + 18, meaning the service gains 3 thousand subscribers per month. The key error to avoid is choosing an exponential model when the problem states "increases by the same amount," which indicates additive (linear) growth, not multiplicative (exponential) growth.
A candle burns down at a constant rate. Its height is 18 cm at $t=0$ minutes and 12 cm at $t=10$ minutes. Assuming a linear model $h(t)=mt+b$ , what is $m$ , and what is the candle's height after 25 minutes?
Answer: $m=-0.6$ ; 3 cm This problem involves a candle burning at a constant rate, indicating linear decay. Using h(t) = mt + b with the given data: at t = 0, h = 18, so b = 18; at t = 10, h = 12. This gives us 12 = 10m + 18, solving to m = -0.6 (negative because height decreases). The model is h(t) = -0.6t + 18. After 25 minutes: h(25) = -0.6(25) + 18 = -15 + 18 = 3 cm. The constant rate of -0.6 cm/minute represents how much height the candle loses each minute. Be careful with the sign of the slope in decay problems - it should be negative.
Solve the system $\begin{cases}5x+y=1\\10x+2y=4\end{cases}$ How many solutions does this system have?
Answer: $0$ We need to determine how many solutions the system $5x + y = 1$ and $10x + 2y = 4$ has. I'll check if the second equation is a multiple of the first. If we multiply the first equation by 2, we get $10x + 2y = 2$ , not $10x + 2y = 4$ . Since the left sides are the same but the right sides differ ( $2 \neq 4$ ), these equations represent parallel lines that never intersect. The system has no solutions ( $0$ solutions). The key insight is that when equations have proportional left sides but non-proportional right sides, the lines are parallel. Always check both sides when looking for multiples.
A rectangle has perimeter 30. If its length is $x$ and its width is $15-x$ , its area is $A(x)=x(15-x)$ . For what value of $x$ is the area maximized?
Answer: $x=7.5$ The area function $A(x)=x(15-x)=15x-x^2$ represents a downward-opening parabola. To find the maximum, we use the vertex formula $x=-\frac{b}{2a}$ where the function is written as $A(x)=-x^2+15x$ . Here $a=-1$ and $b=15$ , so $x=-\frac{15}{2(-1)}=\frac{15}{2}=7.5$ . This makes sense physically: for a rectangle with fixed perimeter, the maximum area occurs when it's a square (length equals width). Common errors include using the wrong formula or not recognizing that this is an optimization problem requiring the vertex.
A student uses the equation $3x+5y=40$ to model a combination of $x$ large boxes and $y$ small boxes that together weigh 40 pounds. Which equation gives $y$ in terms of $x$ ?
Answer: $y=\frac{40-3x}{5}$ This question requires solving the equation 3x + 5y = 40 for y in terms of x, modeling the weight of boxes. Start by isolating the y term: subtract 3x from both sides to get 5y = 40 - 3x, then divide by 5 to obtain y = (40 - 3x)/5. This matches choice A and correctly expresses y as a function of x. A frequent error is swapping the coefficients in the numerator, as in choice B, or using a negative in the numerator like choice C. Choice D might result from incorrectly simplifying or dividing the entire equation. When manipulating equations with two variables, systematically isolate the desired variable by performing inverse operations, ensuring the relationship is preserved.
A company's profit is modeled by $P=50n-200$ , where $n$ is the number of items sold and $P$ is profit in dollars. What is the profit when $n=8$ , and which part of the equation represents a fixed cost?
Answer: $P=200$ , fixed cost $-200$ This question requires calculating profit P when n = 8 in P = 50n - 200 and identifying the fixed cost term. Substitute n = 8: 50 × 8 - 200 = 400 - 200 = 200, so profit is 200 dollars. The constant -200 represents the fixed cost, as it does not depend on n. Choices with P = 600 likely come from adding instead of subtracting 200, a sign error. Other errors might misidentify the coefficient 50 as fixed. When analyzing profit models, distinguish the constant as fixed components and the coefficient as variable per-unit contributions to understand cost-profit dynamics.
In the coordinate plane, points $A(-2,1)$ and $B(4,1)$ form a horizontal segment. Point $C(4,6)$ is directly above $B$ . What is the length of segment $AC$ ?
Answer: $\sqrt{61}$ This question requires calculating the length of segment AC given coordinates A(-2,1) and C(4,6). The distance formula is sqrt[ $(x2 - x1)^2$ + $(y2 - y1)^2$ ], emphasizing the Pythagorean theorem in the coordinate plane. Compute differences: Δx = 4 - (-2) = 6, Δy = 6 - 1 = 5; then sqrt(36 + 25) = sqrt(61). Point B is given but not needed for AC. A common error is forgetting to square the differences or misapplying the formula. Another is confusing coordinates. Plot points mentally to verify the calculation makes sense.
A right triangle has legs $8$ and $15$ . A second right triangle is similar to the first and has a shorter leg of length $24$ . What is the length of the hypotenuse of the second triangle?
Answer: $51$ This problem involves similar triangles where the first has legs 8 and 15, and the second has a shorter leg of 24. First, find the hypotenuse of the original triangle: c² = 8² + 15² = 64 + 225 = 289, so c = 17. The original triangle has sides 8, 15, and 17. Since the shorter leg of the original is 8 and the shorter leg of the similar triangle is 24, the scale factor is 24/8 = 3. Therefore, the hypotenuse of the second triangle is 17 × 3 = 51. A common error is to assume 15 is the shorter leg or to incorrectly apply the scale factor. Always identify which leg corresponds between similar triangles.
A right triangle has legs 9 and 40. What is the measure of the acute angle $\theta$ opposite the leg of length 9, to the nearest degree? Use $\tan^{-1}\left(\frac{\text{opposite}}{\text{adjacent}}\right)$ .
Answer: $13^\circ$ We need to find the angle θ opposite the leg of length 9, with the other leg being 40. Since we know both legs (opposite = 9, adjacent = 40), we use tangent: tan(θ) = opposite/adjacent = 9/40 = 0.225. Using inverse tangent: θ = tan⁻¹(0.225) ≈ 12.68°, which rounds to 13°. A common error is confusing which leg is opposite versus adjacent to the angle being found. When both legs are given, carefully identify which angle you're finding and which leg is opposite to that specific angle.
At 60 miles per hour, what is the speed in feet per second?
Answer: 88 ft/sec We need to convert 60 miles per hour to feet per second, which requires two conversion factors. Set up the conversion using dimensional analysis: 60 mph × (5280 ft/1 mile) × (1 hour/3600 seconds). Multiply to get: 60 × 5280 ÷ 3600 = 316,800 ÷ 3600 = 88 ft/sec. Notice how the units cancel out properly: miles cancel with miles, hours cancel with hours, leaving feet per second. A common error is forgetting that there are 5280 feet in a mile or 3600 seconds in an hour. Remember to always write out the conversion factors as fractions to ensure proper unit cancellation.
A gym charges a one-time sign-up fee plus a monthly membership fee. The total cost $C$ (in dollars) after $m$ months is $C=155$ after 6 months and $C=235$ after 10 months. Which equation represents this relationship in slope-intercept form?
Answer: $C=20m+35$ The question asks for the slope-intercept form of the equation representing the total cost C after m months, given C=155 after 6 months and C=235 after 10 months. To find this, first calculate the slope, which is the monthly fee, using the formula m = (C2 - C1)/(m2 - m1) = (235 - 155)/(10 - 6) = 80/4 = 20. Then, use one point to find the y-intercept: 155 = 20*6 + b, so b = 155 - 120 = 35, giving C = 20m + 35. This equation shows the linear relationship where the slope is the constant monthly rate and the intercept is the sign-up fee. A common computational error is miscalculating the slope by subtracting in the wrong order, but since slope is rise over run, consistency yields the same result. Another error might be forgetting to solve for the intercept after finding the slope. A useful test-taking strategy is to verify the equation with both given points to ensure accuracy.
A designer compares two curves on a coordinate plane: the line $y=-2x+4$ and the parabola $y=x^2-4x+4$ . What are all solutions $(x,y)$ to the system?
Answer: $(0,4)$ and $(2,0)$ We need to solve the system y = -2x + 4 and y = x² - 4x + 4. This combines a linear equation with a quadratic, solved by substitution. Setting equal: -2x + 4 = x² - 4x + 4, which simplifies to 0 = x² - 4x + 2x, or 0 = x² - 2x. Factoring: x(x - 2) = 0, so x = 0 or x = 2. When x = 0, y = -2(0) + 4 = 4; when x = 2, y = -2(2) + 4 = 0. The solutions are (0, 4) and (2, 0). A common error is canceling the +4 incorrectly; always be careful with constant terms. Since we found two intersection points, this confirms the typical case where a parabola and line intersect at exactly two points.
$\sqrt{5x+1}=x-1$ What is the solution to the given equation?
Answer: $7$ Square both sides: $5x+1=(x-1)^2=x^2-2x+1$ . Rearranging gives $x^2-7x=0$ , so $x(x-7)=0$ , meaning $x=0$ or $x=7$ . Since the original equation requires $x-1\ge 0$ (the square root cannot be negative), $x=0$ gives $x-1=-1$ , which is invalid — this is an extraneous root introduced by squaring. Checking $x=7$ : $x-1=6\ge 0$ and $\sqrt{5(7)+1}=\sqrt{36}=6$ , which matches. So $x=7$ is the only valid solution.

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