Digital SAT Math Practice Test 1 Walkthrough Notes

Summary: This note provides a comprehensive walkthrough of the Digital SAT Practice Test 1 math sections, including both Section 1 and Section 2. Each question is analyzed, solutions are explained, and relevant strategies and tips are discussed.

Highlights:

Section 1: Covers various topics including percentages, equation solving, inequalities, function evaluation, probability, unit conversion, similar triangles, systems of equations, linear models, interpreting graphs, factoring, exponential growth, and geometry.
Section 2: Addresses topics such as algebraic expressions, linear functions, systems of equations, interpreting graphs, exponential functions, data analysis, circle equations, absolute value, volume calculations, and quadratic equations.
Strategies and Tips: Emphasizes paying attention to units, understanding translations, recognizing common question types (e.g., stacked equations, exponential growth), and utilizing Desmos for complex graphing problems.

Detailed Walkthrough:

Section 1:

Percentage Calculation: 10% of 470 is calculated as 470 * 0.1 = 47.
Equation Solving: Solving 4x + 6 = 18 yields 4x = 12.
Inequality: The inequality representing the surfboard rental scenario is 25 + 10t ≤ 75.
Function Evaluation: Solving g(x) = x² + 9 = 25 gives x = 4.
Probability: The probability of rolling a 2 on a 14-sided die is 1/14.
Unit Conversion: Converting 42 posters/minute to posters/hour: 42 * 60 = 2520 posters/hour.
Function Evaluation: f(4) = 7(4) + 2 = 30.
Equation Representation: The equation representing the point system for the assignment is x + 3y = 70.
Similar Triangles: Since triangles LMN and PQR are similar and angle M is 53°, angle Q is also 53°.
System of Equations (Substitution): Substituting y = -3x into 4x + y = 15 gives x = 15.
Linear Model: The most appropriate linear model for the scatter plot data has a negative slope and a positive y-intercept.
Graph Interpretation (x-intercepts): The graph of f(x) intersects the x-axis at three points.
Word Problem (Substitution): Solving 3h + c = 71 with h = 10 gives c = 41.
Factoring Quadratic: Factoring z² + 10z - 24 = 0 gives (z+12)(z-2) = 0, so z = 2 is a solution.
Exponential Growth: The number of bacteria after 15 hours is 300,000 * 2^(15/3) = 9,600,000.
Factoring Expression: 6x⁸y² + 12x²y² is equivalent to 6x²y²(x⁶ + 2).
Interpretation in Context: 'x' represents the average number of trees per hectare in the park.
Linear Equation from Graph: The equation representing the relationship between shares of stock is 8x + 12y = 480.
Circle Area Ratio: The area of circle B is (129n)²π / (3n)²π = 1849 times the area of circle A.
Frequency Table Interpretation: The maximum data value in the frequency table is 14.
Circle Diameter: The diameter of the circle with endpoints (2,4) and (2,14) is 14 - 4 = 10, so the radius is 5.
Angle Conversion (Radians to Degrees): Angle T is (2π/3 + 5π/12) * (180/π) = 195 degrees.
Area Conversion (Square Miles to Square Yards): 4.36 square miles * (1760 yards/mile)² = 13,505,536 square yards.
Linear Equation and Translation: The x-intercept of line K (translated down 5 units from line H) is found by solving 0 = 6x + 17, which gives x = -17/6.
Quadratic and Intersection Point (Desmos): The y-coordinate of the vertex of y = -x² + 9x - 100 is -79.75, which is equal to c.
System of Equations: The ordered pair (-3r + 7, r) is a solution to the given system of equations.
Equilateral Triangle and 30-60-90 Triangle: The height of the equilateral triangle is 104√3, so K = 104.

Section 2:

Algebraic Expression: The amount earned for 39W hours of work is 39P.
Linear Function: The function modeling the time to ride x miles is M(x) = 5.7x.
System of Equations: Adding the two equations in the system gives y = 6.
Function Evaluation: The speed after 5 seconds is s(5) = 40 + 3(5) = 55 mph.
Pythagorean Theorem: The value of c is √(4² + 5²) = √41.
Equation Solving: 4x + 5 = 165, so x = 40.
Graph Interpretation (x-intercept): The x-intercept of the graph is 7.
Y-intercept: The y-intercept of f(x) = (1/10)x - 2 is -2.
Function Transformation (Vertical Shift): g(x) = 7x³ - 2.
System of Equations (Substitution): If x + 7 = 10, then y = (x+7)² = 10² = 100.
Simplifying Expression: 7x³ + 7x - 6x³ - 3x simplifies to x³ + 10x.
Function Evaluation: Solving 56 = 7n³ gives n = 2.
Parallel Lines and Angles: x = 180 - 110 = 70°.
Mean Calculation: The mean of the data set is 100/10 = 10.
Exponential Function Interpretation: The number 5 represents the estimated number of employees when the restaurant opened.
Minimum Value of a Function: The minimum value of x² + 55 is 55.
Function Type (Exponential vs. Linear): The function best modeling the investment value increase is increasing exponential.
Percentage Increase: K = 1.07.
Exponent Rules: a^(11/12) is equivalent to the 144th root of a^132.
Word Problem (Inequality): The greatest number of attendees without exceeding the budget is 16.
Absolute Value Equation: Solving |4x - 4| = 112 gives x = 29. The positive value of x - 1 is 28.
Volume Calculation: The volume of the space not taken up by the sphere is 68³ - (4/3)π(34³) ≈ 149796 cubic inches.
Circle Equation (Diameter): The diameter of the circle is 2 * √16 = 8.
Exponential Function (Coefficient/Base): The function f(t) = 128(1.6)^(t-1) shows K = 128 as a coefficient.
Exponential Growth (Word Problem): The equation representing the squirrel population model is n = 30 + 150t.
Perpendicular Lines: The pair of equations representing perpendicular lines is 7x + 10y = 5 and -7x + 5y = 2.
Quadratic Equations (Discriminant): If the equation x² - 34x + c = 0 has no real solutions, then (-34)² - 4(1)(c) < 0, which implies c > 289. The least possible integer value for n is 289.