NJGPA - New Jersey Graduation Proficiency Assessment Algebra I and Geometry Concepts Questions and Answers 1 — Questions and Answers
Question 1: A local gym charges a one-time registration fee and a monthly membership fee. The table below shows the total cost (y) for a certain number of months (x). | Months (x) | Total Cost (y) | |------------|----------------| | 2 | $130 | | 4 | $210 | | 6 | $290 | Which of the following linear equations models the total cost for x months of membership?
- y = 40x + 50 (Correct answer)
- y = 50x + 30
- y = 80x - 30
- y = 45x + 40
Correct answer: y = 40x + 50
To find the linear equation y = mx + b, first find the slope (m), which is the monthly fee. Using two points from the table, (2, 130) and (4, 210), the slope m = (210 - 130) / (4 - 2) = 80 / 2 = $40 per month. Now, substitute the slope and one of the points into the slope-intercept form to find the y-intercept (b), which is the registration fee. Using (2, 130): 130 = 40(2) + b. 130 = 80 + b. b = 50. Therefore, the equation is y = 40x + 50.
Question 2: A rectangular garden has a length of (x + 7) feet and a width of (x - 2) feet. If the area of the garden is 50 square feet, which equation can be used to find the value of x?
- x² + 5x - 14 = 50 (Correct answer)
- x² - 14x + 5 = 50
- 2x + 5 = 50
- x² + 9x - 14 = 50
Correct answer: x² + 5x - 14 = 50
The area of a rectangle is calculated by multiplying its length by its width. In this scenario, the area is (x + 7)(x - 2). To set up the equation, expand the binomials: (x + 7)(x - 2) = x(x) + x(-2) + 7(x) + 7(-2) = x² - 2x + 7x - 14 = x² + 5x - 14. Since the area is given as 50 square feet, the resulting equation is x² + 5x - 14 = 50.
Question 3: In the figure below, line m is parallel to line n, and they are intersected by transversal line t. If the measure of angle 2 is 115°, what is the measure of angle 7?
- 115°
- 65° (Correct answer)
- 75°
- Cannot be determined
Correct answer: 65°
Angle 2 and angle 6 are corresponding angles, so they are congruent; m∠6 = 115°. Angle 6 and angle 7 are supplementary angles because they form a linear pair. Therefore, their sum is 180°. To find the measure of angle 7, subtract the measure of angle 6 from 180°: 180° - 115° = 65°. Alternatively, Angle 2 and Angle 3 are supplementary (180 - 115 = 65). Angle 3 and Angle 7 are alternate interior angles, so they are congruent. Thus, m∠7 = 65°.
Question 4: A city park is shaped like a right triangle. The two legs of the triangle measure 8 miles and 15 miles. A new path is being built along the hypotenuse of the triangle. What is the length of the new path?
- 23 miles
- sqrt(161) miles
- 17 miles (Correct answer)
- 161 miles
Correct answer: 17 miles
This problem requires the use of the Pythagorean theorem, a² + b² = c², where 'a' and 'b' are the legs of the right triangle and 'c' is the hypotenuse. Substitute the given values: 8² + 15² = c². Calculate the squares: 64 + 225 = c². Add the results: 289 = c². Finally, find the square root of 289 to solve for c: c = √289 = 17. The length of the path is 17 miles.
Question 5: Which of the following are the solutions to the quadratic equation x² - 3x - 10 = 0?
- x = -5 and x = 2
- x = 5 and x = -2 (Correct answer)
- x = 10 and x = -1
- x = -10 and x = 1
Correct answer: x = 5 and x = -2
To solve the quadratic equation, we need to factor the trinomial x² - 3x - 10. We are looking for two numbers that multiply to -10 and add up to -3. These numbers are +2 and -5. Therefore, the factored form is (x + 2)(x - 5) = 0. According to the zero-product property, we set each factor equal to zero to find the solutions. So, x + 2 = 0 gives x = -2, and x - 5 = 0 gives x = 5.
Question 6: A homeowner is calculating the area of a room that has an L-shape. The composite shape can be divided into two rectangles. The first rectangle measures 10 feet by 12 feet. The second, smaller rectangle measures 6 feet by 7 feet. What is the total area of the room?
- 194 square feet
- 120 square feet
- 42 square feet
- 162 square feet (Correct answer)
Correct answer: 162 square feet
To find the area of a composite shape, you must break it down into simpler shapes, find the area of each part, and then add them together. The area of the first rectangle is length × width = 10 ft × 12 ft = 120 square feet. The area of the second rectangle is 6 ft × 7 ft = 42 square feet. The total area is the sum of the areas of the two rectangles: 120 sq ft + 42 sq ft = 162 square feet.
NJGPA - New Jersey Graduation Proficiency Assessment Algebra I and Geometry Concepts Questions and Answers 1
A local gym charges a one-time registration fee and a monthly membership fee.
The table below shows the total cost (y) for a certain number of months (x).
| Months (x) | Total Cost (y) |
|------------|----------------|
| 2 | $130 |
| 4 | $210 |
| 6 | $290 |
Which of the following linear equations models the total cost for x months of membership?