A student is asked to solve the equation √(x + 7) - 1 = x. The student provides the following work: Step 1: √(x + 7) = x + 1 Step 2: x + 7 = (x + 1)² Step 3: x + 7 = x² + 2x + 1 Step 4: 0 = x² + x - 6 Step 5: 0 = (x + 3)(x - 2) Step 6: x = -3 or x = 2 Which of the following evaluations of the student's work is most accurate and provides the best pedagogical feedback?
The population of a species of fish in a lake is modeled by the function P(t) = 2500e^(0.05t), where t is the number of years since 2020. A biologist determines that the rate of change of the fish population is 150 fish per year. At what year will this occur?
A city park is shaped like a composite figure formed by a rectangle and a semicircle. The rectangle has a length of 100 meters and a width of 60 meters. The diameter of the semicircle is the 60-meter width of the rectangle. A groundskeeper needs to install a fence around the entire perimeter of the park. What is the total length of fencing required, rounded to the nearest meter?
A high school statistics class collected data on the number of hours students study per week and their corresponding GPA. They calculated a linear regression model: `GPA = 2.5 + 0.08 * (Hours)`. Which TWO of the following statements are correct interpretations of this model? (Select TWO)
True or False: For any two distinct irrational numbers, their sum must also be an irrational number.
A manufacturing firm is designing a cylindrical can that must hold 500 cubic centimeters of liquid. The material for the circular top and bottom of the can costs $0.06 per square centimeter, while the material for the side of the can costs $0.03 per square centimeter. The firm wants to minimize the cost of materials for each can. The cost function for the can is C(r) = 0.12πr² + 30/r, where r is the radius of the can in centimeters. What is the approximate radius that minimizes the cost of the materials for one can?
The vertices of triangle ABC are located at A(1, 2), B(7, 2), and C(4, 6). The triangle is first reflected across the x-axis to create triangle A'B'C'. Then, triangle A'B'C' is translated 3 units to the left and 1 unit up to create triangle A''B''C''. What are the coordinates of vertex C''?
The expression (3x³ - 5x² + x + 1) is divided by (x - 2). According to the Remainder Theorem, the remainder of this division is equal to the value of the polynomial when x = ____.
A probability distribution for a discrete random variable X is given in the table below. | X | P(X) | |---|---| | 0 | 0.1 | | 1 | 0.3 | | 2 | 0.4 | | 3 | ? | What is the expected value, E(X), of the random variable?
Consider the function f(x) = |x - 3| + 2. Which of the following statements accurately describes the continuity and differentiability of this function at x = 3?