The correct answer is 1.25 or ⁵⁄₄. The y-coordinate of the x-intercept is 0, so 0 can be substituted for y, giving ⁴⁄₅x + ¹⁄₃(0) = 1. This simplifies to ⁴⁄₅ x = 1. Multiplying both sides of ⁴⁄₅x = 1 by 5 gives 4x = 5. Dividing both sides of 4x = 5 by 4 gives x = ⁵⁄₄, which is equivalent to 1.25. Either 5/4 or 1.25 may be gridded as the correct answer

The correct answer is 5. The intersection points of the graphs of y = 3x²− 14x and y = x can be found by solving the system consisting of these two equations. To solve the system, substitute x for y in the first equation. This gives x = 3x² − 14x. Subtracting x from both sides of the equation gives 0 = 3x² − 15x. Factoring 3x out of each term on the left-hand side of the equation gives 0 = 3x(x − 5). Therefore, the possible values for x are 0 and 5. Since y = x, the two intersection points are (0, 0) and (5, 5). Therefore, a = 5.

The possible correct answers are 97, 98, 99, 100, and 101. The volume of a cylinder can be found by using the formula V = πr²h, where r is the radius of the circular base and h is the height of the cylinder. The smallest possible volume, in cubic inches, of a graduated cylinder produced by the laboratory supply company can be found by substituting 2 for r and 7.75 for h, giving V = π(2²)(7.75). This gives a volume of approximately 97.39 cubic inches, which rounds to 97 cubic inches. The largest possible volume, in cubic inches, can be found by substituting 2 for r and 8 for h, giving V = π(2²)(8). This gives a volume of approximately 100.53 cubic inches, which rounds to 101 cubic inches. Therefore, the possible volumes are all the integers greater than or equal to 97 and less than or equal to 101, which are 97, 98, 99, 100, and 101. Any of these numbers may be gridded as the correct answer.

The correct answer is 31. The equation can be solved using the steps shown below.

2(5x − 20) − 15 − 8x = 7

2(5x) − 2(20) − 15 − 8x = 7 (Apply the distributive property.)

10x − 40 − 15 − 8x = 7 (Multiply.)

2x − 55 = 7 (Combine like terms.)

2x = 62 (Add 55 to both sides of the equation.)

x = 31 (Divide both sides of the equation by 2.)

The correct answer is 10.Let n be the number of friends originally in the group. Since the cost of the trip was $800, the share, in dollars, for each friend was originally \(\frac{800}{n}\). When two friends decided not to go on the trip, the number of friends who split the $800 cost became n − 2, and each friend’s cost became \(\frac{800}{n-2}\). Since this share represented a $20 increase over the original share, the equation \(\frac{800}{n}+20=\frac{800}{n-2}\) must be true. Multiplying each side of \(\frac{800}{n}+20=\frac{800}{n-2}\) by n(n − 2) to clear all the denominators gives

800(n − 2) + 20n(n − 2) = 800n

This is a quadratic equation and can be rewritten in the standard form by expanding, simplifying, and then collecting like terms on one side, as shown below:

800n − 1600 + 20n² − 40n = 800n

40n − 80 + n²− 2n = 40n

n² − 2n − 80 = 0

After factoring, this becomes (n + 8)(n − 10) = 0.

The solutions of this equation are −8 and 10. Since a negative solution makes no sense for the number of people in a group, the number of friends originally in the group was 10.

The data in the scatterplot roughly fall in the shape of a downward-opening parabola; therefore, the coefficient for the x² term must be negative. Based on the location of the data points, the y-intercept of the parabola should be somewhere between 740 and 760. Therefore, of the equations given, the best model is y = −1.674x² + 19.76x + 745.73.

Choices A and C are incorrect. The positive coefficient of the x² term means that these these equations each define upward-opening parabolas, whereas a parabola that fits the data in the scatterplot must open downward. Choice B is incorrect because it defines a parabola with a yintercept that has a negative y-coordinate, whereas a parabola that fits the data in the scatterplot must have a y-intercept with a positive y-coordinate.

The average speed of the model car is found by dividing the total distance traveled by the car by the total time the car traveled. In the first t seconds after the car starts, the time changes from 0 to t seconds. So the total distance the car traveled is the distance it traveled at t seconds minus the distance it traveled at 0 seconds. At 0 seconds, the car has traveled 16(0)√0 inches, which is equal to 0 inches. According to the equation given, after t seconds, the car has traveled 16t√t inches. In other words, after the car starts, it travels a total of 16t√t inches in t seconds. Dividing this total distance traveled by the total time shows the car’s average speed: \(\frac{16t\sqrt{t}}{t}\) = 16t√t inches per second.

Choices A, C, and D are incorrect and may result from misconceptions about how average speed is calculated.

The coordinates of the points at a distance d units from the point with coordinate a on the number line are the solutions to the equation |x − a| = d. Therefore, the coordinates of the points at a distance of 3 units from the point with coordinate −4 on the number line are the solutions to the equation |x − (−4)| = 3, which is equivalent to |x + 4| = 3.

Choice B is incorrect. The solutions of |x − 4| = 3 are the coordinates of the points on the number line at a distance of 3 units from the point with coordinate 4. Choice C is incorrect. The solutions of |x + 3| = 4 are the coordinates of the points on the number line at a distance of 4 units from the point with coordinate −3. Choice D is incorrect. The solutions of |x − 3| = 4 are the coordinates of the points on the number line at a distance of 4 units from the point with coordinate 3.

One way to find the radius of the circle is to put the given equation in standard form, (x − h)² + (y − k)² = r², where (h, k) is the center of the circle and the radius of the circle is r. To do this, divide the original equation, 2x² − 6x + 2y² + 2y = 45, by 2 to make the leading coefficients of x² and y² each equal to 1: x² − 3x + y² + y = 22.5. Then complete the square to put the equation in standard form. To do so, first rewrite x² − 3x + y² + y = 22.5 as (x² − 3x + 2.25) − 2.25 + (y² + y + 0.25) − 0.25 = 22.5. Second, add 2.25 and 0.25 to both sides of the equation: (x² − 3x + 2.25) + (y² + y + 0.25) = 25. Since x² − 3x + 2.25 = (x − 1.5)², y² − x + 0.25 = (y − 0.5)² , and 25 = 5² , it follows that (x − 1.5)² + (y − 0.5)² = 5². Therefore, the radius of the circle is 5.

Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.

Since Gear A has 20 teeth and Gear B has 60 teeth, the gear ratio for Gears A and B is 20:60. Thus the ratio of the number of revolutions per minute (rpm) for the two gears is 60:20, or 3:1. That is, when Gear A turns at 3 rpm, Gear B turns at 1 rpm. Similarly,since Gear B has 60 teeth and Gear C has 10 teeth, the gear ratio for Gears B and C is 60:10, and the ratio of the rpms for the two gears is 10:60. That is, when Gear B turns at 1 rpm, Gear C turns at 6 rpm. Therefore, if Gear A turns at 100 rpm, then Gear B turns at \(\frac{100}{3}\) rpm, and Gear C turns at \(\frac{100}{3}\) × 6 = 200 rpm.

Alternate approach: Gear A and Gear C can be considered as directly connected since their “contact” speeds are the same. Gear A has twice as many teeth as Gear C, and since the ratios of the number of teeth are equal to the reverse of the ratios of rotation speeds, in rpm, Gear C would be rotated at a rate that is twice the rate of Gear A. Therefore, Gear C will be rotated at a rate of 200 rpm since Gear A is rotated at 100 rpm.

Choice A is incorrect and may result from using the gear ratio instead of the ratio of the rpm when calculating the rotational speed of Gear C. Choice B is incorrect and may result from comparing the rpm of the gears using addition instead of multiplication. Choice D is incorrect and may be the result of multiplying the 100 rpm for Gear A by the number of teeth in Gear C.