Math: Equations, Inequalities and Probability

Question 1: Percentage Calculation

• To find 10% of 470, multiply 470 by 0.1.
• The result of the calculation is 47.

Question 2: Equation Solving

• The goal is to find an equation with the same solution as the given equation.
• Subtract 6 from both sides of the original equation to isolate the term with 'x'.
• The simplified equation is 4x = 12.

Question 3: Inequality Representation

• The total cost consists of a $25 service fee and a $10 per hour rental fee.
• The person intends to spend a maximum of $75 to rent the surfboard for 'T' hours.
• The inequality representing this situation is 25 + 10T ≤ 75.

Question 4: Function Evaluation

• The function G is defined as G(x) = x² + 9.
• Set G(x) equal to 25 to find the value of x for which G(x) = 25.
• The equation becomes 25 = x² + 9.
• Solving for x, we find that x = 4.

Question 5: Probability Calculation

• A fair 14-sided die is labeled with numbers 1 through 14.
• The probability of rolling a 2 is 1 out of 14, since there is only one face with the number 2.

Question 6: Unit Conversion

• A printer produces posters at a constant rate of 42 posters per minute.
• Convert this rate to posters per hour.
• Multiply 42 posters per minute by 60 minutes per hour.
• The printer produces 2520 posters per hour.

Question 7: Function Evaluation

• The function f is defined by the equation f(x) = 7x + 2.
• To find the value of f(x) when x = 4, substitute 4 for x in the equation.
• f(4) = 7(4) + 2 = 28 + 2 = 30.

Question 8: Equation Representation

• An assignment is worth 70 points and consists of one-point and three-point questions.
• 'X' represents the number of one-point questions and 'y' represents the number of three-point questions.
• The equation representing this situation is x + 3y = 70.

Question 9: Similar Triangles

• Right triangles LMN and PQR are similar.
• Angle M has a measure of 53 degrees.
• The measure of angle Q is also 53 degrees, since angle M corresponds to angle Q in similar triangles.

Question 10: System of Equations

• The solution to the given system of equations is (x, y).
• The goal is to find the value of x.
• Given y = -3x, substitute -3x for y in the other equation.
• The equation becomes 4x + (-3x) = 15, which simplifies to x = 15.

Question 11: Linear Modeling

• Find the most appropriate linear model for the data shown in the scatter plot.
• The y-intercept is approximately 10 and the slope is negative.
• Option B has the positive Y intercept and the negative slope

Question 12: Graph Interpretation

• Given a graph y = f(x), where f(x) = ax³ + bx² + cx + d.
• Determine for how many values of x does f(x) equal zero.
• Count the number of x-intercepts on the graph, which occur at three different points.

Question 13: System of Equations

• Vivian buys party hats and cupcakes for $71.
• Each package of party hats costs $3, and each cupcake costs $1.
• X represents party hats and y represents cupcakes.
• Vivian bought 10 packages of party hats.
• The equation is 3h + c = 71, where H is 10
• The solution is C = 41

Question 14: Solving Quadratic Equations

• Find one of the solutions to the equation z² + 10z - 24 = 0.
• Factor the quadratic equation to find the values of z that make the equation equal to zero.
• The factored form is (z + 12)(z - 2) = 0.
• One solution is z = 2.

Question 15: Exponential Growth

• Bacteria are growing in a liquid growth medium.
• There were initially 300,000 cells per milliliter.
• The number of cells per milliliter doubles every 3 hours.
• Determine the number of cells per milliliter 15 hours after the initial observation.
• The formula for exponential growth is Initial Value * (Growth Rate)^(Time/Doubling Time)
• The equation is 300,000 * 2^(15/3)
• Solved, this is 9.6 million.

Question 16: Equivalent Expressions

• Find the expression that is equivalent to 6x⁸y² + 12x²y².
• Factor out the greatest common factor from both terms.
• The greatest common factor is 6x²y².
• Factoring out 6x²y² from both terms, the expression becomes 6x²y²(x⁶ + 2).

Question 17: Interpretation of Variables

• A neighborhood consists of a 2-hectare park and a 35-hectare residential area.
• The total number of trees in the neighborhood is 3934.
• The equation 2x + 35y = 3934 represents the situation.
• Determine the interpretation of x in the context.
• X is the average number of trees per hectare in the park.

Question 18: Equation Representation

• The graph shows the relationship between the number of shares of stock from Company A (x) and Company B (y) that Simone can purchase.
• Determine which equation can represent this relationship.
• y intercepts are around 40 so there is likely a standard linear form.
• The value of slope M is equivalent to -A/B and A is what's in front of the X value and B is what's in front of the Y value### Slope Calculation and Analysis
• Down 40 over 60 simplifies to -2/3.
• Glancing at answer choice D immediately reveals it's incorrect due to flipped coefficients and same signs.
• Calculation of slope for answer D is -12/8, simplifying to -3/2, which is not the correct slope.

Circle Area Ratio

• Circle A has a radius of 3n, while Circle B has a radius of 129n.
• Area of a circle is calculated using the formula πr².
• The ratio of Circle B's area to Circle A's area is (π(129n)²) / (π(3n)²).
• The π terms cancel out, simplifying the ratio.
• The ratio is further simplified to (129² * n²) / (3² * n²).
• The n² terms also cancel out, leaving 129² / 3².
• Calculation yields 16641 / 9 = 1849.

Maximum Data Value

• The question asks for the maximum data value in the data set, not the data value with the highest frequency.
• The maximum data value in the given data set is 14.
• The data value with the highest frequency is 11, but that is not what the question is asking for.

Circle and XY Plane

• The circle in the XY plane has a diameter with endpoints (2, 4) and (2, 14).
• The X values of the end points are the same, so the diameter is the different of the Y values..
• The diameter is 14 - 4 = 10.
• The radius is half of the diameter, so radius = 5.
• The equation of the circle is (x - 2)² + (y - 9)² = r², where r is a positive constant.
• The value of r is 5.
• Pay attention to whether X or Y values are the same.

Angle Conversion

• Angle T is equal to angle R plus 5π/12 radians.
• Angle R is 2π/3 radians.
• Common denominator of 12 leads to the equation where angle T is equal to 8π/12 + 5π/12.
• This simplifies to 13π/12.
• To convert radians to degrees, multiply by 180 degrees per π radians.
• After canceling out radians, the equation is 13/12 * 180 degrees.
• Angle T is 195 degrees.

Area Conversion

• The town has an area of 4.36 square miles.
• 1 mile equals 1760 yards.
• Convert square miles to square yards the fact that you are squaring the conversion numbers.
• 4.36 square miles is multiplied by 1760² to obtain the area in square yards.
• 1760² multiplied by 4.36 equals 13,505,536 square yards.

Linear Translations

• Line K is a translation of line H down five units in the XY plane.
• Pay attention to translations on the SAT Math section; they are fairly common.
• Calculate slope for Line H by analyzing the change in given x and y values to be 6.
• Calculate the equation by using data points in the table.
• Use the x value 18 to calculate a slope of 6, the solve for the y intercept.
• The y-intercept is then 130 - 108 = 22.
• The translation brings the y-intercept down 5 bringing the new intercept to 17.
• Set the Y values to all zero to solve for where x is y intercept.
• Setting zero to 6x + 17 shows us that x = -17/6

Quadratic Functions

• The equations y = -x² + 9x - 100 and y = c intersect at exactly one point.
• Use Desmos to quickly visualize this answer.
• Graph y = -x² + 9x - 100, and see that it will only bisect at the vertex.
• After clicking on the lowest vertex, the calculator shows an intersection at only one point with a Y value of -79.75
• Converting the answers to decimals shows that the answer must be C, -319/4 = -79.75

Real Number Equations

• Equations are 2x + 3y = 7 and 10x + 15y = 35.
• The equations are multiples of each other.
• This becomes an easy substitution problem.
• Because of the 2x and 3y, prioritize Y answers over 3 and X answers over 2.
• This means testing Option B and C before A and D.
• Testing X with: x = (-3r + 7) / 2 and y = r shows the right answer.
• Substitute back in to show 7 = 7, meaning that the answer is correct.

Equilateral Triangles

• Perimeter of an equilateral triangle is 624 cm.
• The height of the triangle is K * √3 cm.
• Equilateral triangles have equal sides and angles, a key property.
• Drawing out the triangle is helpful, particularly with equal sides and angles.
• Creating right angles at the side of the triangle with the height is helpful.
• Split the triangles to form two 30, 60, and 90 triangles.
• Properties of a 30, 60, and 90 triangle have values x, x * √3,2x (for opposing sides).
• K * √3 is across from the 60 degree side.

Solving for K in an Equilateral Triangle

• Across from the 30-degree angle is represented by K in a 30-60-90 triangle.
• The hypotenuse of the same triangle is 2K.
• In an equilateral triangle, since all sides are the same, each side = 2K.
• The perimeter of the equilateral triangle is 2K + 2K + 2K = 6K.
• Given a perimeter of 624, 6K = 624.
• Dividing both sides by 6, K = 104.

Problem 1: Earnings Calculation

• Sharon earns P dollars for every W hours of work.
• To calculate total earnings for 39W hours, multiply the rate by the time: P/W * 39W.
• The W's cancel, leaving 39P as the total earnings.

Problem 2: Biking Time

• Person one bikes at an average rate of 5.7 minutes per mile.
• To model the time it takes to ride X miles, multiply the rate (minutes per mile) by the number of miles: 5.7 * X
• M(x) = 5.7x models the total time in minutes.

Problem 3: Solving System of Equations

• Given a system of equations, identify that you need to solve for y.
• If 3x and -3x jump out at you, then adding the equations will eliminate x.
• Adding the equations results in y = 6.

Problem 4: Speed After Acceleration

• Given an equation for speed (miles per hour) after T seconds of acceleration.
• Substitute T = 5 seconds into the equation.
• Speed = 40 + 3(5) = 40 + 15 = 55 miles per hour.

Problem 5: Right Triangle Side Length

• Given a right triangle with sides a = 4 and b = 5, solve for C.
• Apply the Pythagorean theorem: a² + b² = c².
• To find C, take the square root of both sides: C = √(4² + 5²).

Problem 6: Solving Linear Equation

• Find the solution to 4x + 5 = 165.
• Subtract 5 from both sides: 4x = 160.
• Divide both sides by 4: x = 40.

Problem 7: X-Intercept of a Graph

• The x-intercept is where the graph crosses the x-axis, meaning y = 0.
• From the graph, when y is zero, the value of x is 7.

Problem 8: Y-Intercept of a Function

• Given f(x) = (1/10)x - 2, find the y-intercept.
• The y-intercept occurs where x = 0.
• f(0) = (1/10)(0) - 2 = -2.
• The y-intercept is (0, -2).

Problem 9: Shifting a Graph

• f(x) = 7x³
• g(x) is the result of shifting f(x) down two units.
• g(x) = 7x³ - 2.

Problem 10: Ordered Pair Solution

• Solve the system: x + 7 = 10 and x + 7² = y.
• Substitute 10 for x + 7 in the second equation: 10² = y.
• y = 100. Only one solution with that y value exists.

Problem 11: Equivalent Expression

• Simplify 7x³ + 7x - 6x³ - (-3x).
• Combine like terms: (7x³ - 6x³) + (7x + 3x) = x³ + 10x.

Problem 12: Solving for N

• Given p(n) = 7n³ and p(n) = 56, solve for n.
• 56 = 7n³
• Divide by 7: n³ = 8.
• Take the cube root: n = 2.

Problem 13: Intersecting Parallel Lines

• When parallel lines are intersected, corresponding angles are equal.
• one angle is 110 degrees, the corresponding angle is also 110 degrees.
• Supplementary angles add up to 180 degrees - which is the straightline angle formed.
• x = 180 - 110 = 70 degrees.

Problem 14: Calculating the Mean

• To find the mean, sum the data values and divide by the number of values.
• The provided data sums to 100.
• There are 10 data values.
• Mean = 100 / 10 = 10.

Problem 15: Interpreting Exponential Functions

• Given E(t) = 5 * (1.8)^t.
• 5 represents the initial value.
• In this context, 5 is the estimated number of employees when the restaurant first opened.

Problem 16: Minimum Value of a Function

• Given x² + 55, determine the minimum value.
• x² is always non-negative (zero or positive).
• The minimum value of x² is 0, so the minimum value of the function is 0 + 55 = 55.

Problem 17: Modeling Investment Growth

• The value of an investment increases by 0.49 of its value each year.
• Note it is an exponential increase and not linear.

Problem 18: Population Increase

• The population of Greenville increased by 7% from 2015 to 2016.
• The 2016 population is K times the 2015 population.
• To represent a 7% increase, multiply the 2015 population by 1.07.
• K = 1.07.

Problem 19: Equivalent Exponential Expression

• Given a^(11/12).
• Rewrite with a numerator of 132:
• a^(11/12) = a^(132/144).
• Rewrite as a root: 144th root of a^132.

Problem 20: Event Planning Budget

• The cost is a $35 venue rental + $10.25 per attendee.
• The budget is $200.
• 35 + 10.25a less than or equal to 200
• 10.25a less than or equal to 165
• Attendees less than or equal to 16.09, so the greatest number of attendees is 16.

Problem 21: Absolute Value Equation

• Solve the absolute value |4x - 4| = 112 for the positive value of x - 1.
• Set it to the following 4x - 4 = 112
• Solve for x: x = 29.
• Find x - 1: 29 - 1 = 28.

Volume of Space in Cube Not Taken Up by Sphere

• The problem involves finding the volume of the space inside a cube that isn't occupied by a sphere contained within it.
• This requires subtracting the volume of the sphere from the volume of the cube.
• The cube has an edge length of 68 inches.
• The sphere has a radius of 34 inches.
• Volume of the cube calculation is 68 cubed.
• Volume of the sphere to calculate is four thirds pi R cubed using 34 as the radius.
• Calculation: 68 cubed minus (4/3) * pi * 34 cubed equals 149796.
• The volume of the space not taken up by the sphere is 149796 cubic inches.

Diameter of a Circle in the X Y Plane

• The problem asks to find the diameter of a circle given its equation in the X Y plane.
• The general equation of a circle is (x - h) squared + (y - k) squared = r squared with center (h, k) and radius r.
• Given equation: x minus 3 squared + y plus 1 squared = 16
• 16 is equal to r squared.
• The radius (r) is the square root of 16 which is 4.
• The diameter is two times the radius.
• The diameter of the circle is 8.

Exponential Function

• Determine which equivalent form of exponential function f shows the value of K as a coefficient or base.
• F of 1 = k
• The goal is to see which equation results in K being a coefficient or base when 1 is plugged in.
• Option C: 128 * 1.6 to the power of (1 - 1).
• 1. 6 to the power of 0 equals 1.
• 128 * 1 = 128.
• Result is 128 which is a coefficient meaning C is the correct answer.

Squirrel Population Model

• The problem involves an exponential model for estimating the number of squirrels.
• From 2015 to 2020, the number of squirrels increased by 150 each year.
• At the end of 2016, there were 180 squirrels.
• N represents the estimated number of squirrels T years after the end of 2015.
• T is less than or equal to 5.
• Need to backtrack to 2015 to get the initial value as the model starts from 2015 not 2016.
• Subtract 150 from the 2016 number (180 squirrels) to find the number in 2015 = 30.
• Identify that a growth factor of 1.5 is a 50% increase, and a growth factor of 2.5 is a 150% increase.

Perpendicular Lines

• The problem involves pairs of equations representing perpendicular lines.
• Given pair of equations is perpendicular.
• Perpendicular lines have slopes that are reciprocals with opposite signs.
• The equations are in standard linear form (Ax + By = C).
• Slope in standard linear form is calculated as m = -A/B.
• If there's two sign changes we're not actually going to end up changing sign in the slope equation there will be one sign change
• Check each pair of choice equations to see if the reciprocals with opposite signs match.
• Only B has has the opposite coefficients and a sign change.

Real Solutions

• The number of solutions is determined by the discriminant, b squared- 4ac.
• If b squared - 4ac is greater than 0, there are two real solutions.
• If b squared - 4ac equals 0, there is one real solution.
• If b squared - 4ac is less than 0, there are no real solutions.
• The problem states that there are no real solutions meaning the discriminant is less than zero.
• In the given equation, a = 1, b = -34, and c = C.
• Substitute a, b, and c into the discriminant: (-34) squared - 4(1)(C) < 0.
• To isolate C, rearrange the inequality: 1156 - 4C < 0.
• Further, 1156 < 4C.
• C must be greater than 289.
• In the given equation C is greater than n.
• If C > n, and C > 289, then the least possible value of n is 289.