Square PQRS, with a side length of 5 units, will be translated 2 units to the right and 2 units up in the standard (x, y) coordinate plane. What is the area, in square units, of the image of PQRS?
- A. 20
- B. 25
- C. 40
- D. 50
- E. 100
Correct Answer & Rationale
Correct Answer: B
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
Other Related Questions
How many solutions does the equation 3x + 9 = 3x - 12 have?
- B. 1
- C. 2
- D. 3
- E. Infinitely many
Correct Answer & Rationale
Correct Answer: A
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.
To determine the number of solutions for the equation 3x + 9 = 3x - 12, we can simplify both sides. Subtracting 3x from each side results in 9 = -12, which is a false statement. Since the equation leads to a contradiction, it indicates that there are no values of x that can satisfy it. Option B (1 solution) suggests a single value exists, which is incorrect. Option C (2 solutions) and D (3 solutions) imply multiple valid values, which is also false. Option E (infinitely many solutions) suggests that any x would satisfy the equation, which is not true given the contradiction. Thus, the equation has no solutions.
sqrt(45) is between what two consecutive whole numbers?
- A. 4 and 5
- B. 5 and 6
- C. 6 and 7
- D. 14 and 15
- E. 22 and 23
Correct Answer & Rationale
Correct Answer: C
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
To determine between which two consecutive whole numbers \(\sqrt{45}\) lies, we can evaluate the squares of whole numbers around it. Calculating, \(6^2 = 36\) and \(7^2 = 49\). Since \(36 < 45 < 49\), it follows that \(6 < \sqrt{45} < 7\). Therefore, \(\sqrt{45}\) is between 6 and 7. Option A (4 and 5) is incorrect as \(4^2 = 16\) and \(5^2 = 25\), which are both less than 45. Option B (5 and 6) is also wrong since \(5^2 = 25\) and \(6^2 = 36\) are still below 45. Option D (14 and 15) and Option E (22 and 23) are far too high, as \(14^2 = 196\) and \(22^2 = 484\) exceed 45.
Emma measured the height of her laptop screen. She reported the height as 8 inches, accurate to the nearest inch. The actual height of the screen must be:
- A. at least 7.5 inches and less than 8.5 inches
- B. at least 7.9 inches and less than 8.1 inches
- C. at least 7.99 inches and less than 8.01 inches
- D. at least 8 inches
- E. exactly 8 inches
Correct Answer & Rationale
Correct Answer: A
When measuring to the nearest inch, values can range from halfway to the next whole number. For Emma's reported height of 8 inches, this means the actual height must be at least 7.5 inches (inclusive) and less than 8.5 inches (exclusive). Option B is too narrow, only allowing for heights between 7.9 and 8.1 inches, which does not encompass all possible values. Option C is even more restrictive, only allowing for heights between 7.99 and 8.01 inches, excluding valid measurements. Option D is incorrect as it suggests the height must be 8 inches or more, which is too limiting. Option E incorrectly states the height must be exactly 8 inches, disregarding the range of possible values.
When measuring to the nearest inch, values can range from halfway to the next whole number. For Emma's reported height of 8 inches, this means the actual height must be at least 7.5 inches (inclusive) and less than 8.5 inches (exclusive). Option B is too narrow, only allowing for heights between 7.9 and 8.1 inches, which does not encompass all possible values. Option C is even more restrictive, only allowing for heights between 7.99 and 8.01 inches, excluding valid measurements. Option D is incorrect as it suggests the height must be 8 inches or more, which is too limiting. Option E incorrectly states the height must be exactly 8 inches, disregarding the range of possible values.
When Henry plays the songs on the playlist in a random order, what is the probability a rock song will be played first?
- A. 3/4
- B. 1/3
- C. 1/4
- D. 3/10
- E. 5/16
Correct Answer & Rationale
Correct Answer: D
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.