Elementary Literacy & Education — DZ01 Mathematics for Elementary Educators III MATH 1330 Version 1
1. A circle has a radius of 4.5 cm. What is the area of the circle?
Answer: B
The area of the circle is 31.79 cm squared.
To find the area of a circle, the formula \( A = \pi r^2 \) is used. With a radius of 4.5 cm, the area calculates to approximately 31.79 cm squared.
A) 28.26 cm squared
This option is incorrect because the area calculated using the formula does not yield this result. Specifically, when applying the radius of 4.5 cm in the formula \( A = \pi (4.5)^2 \), the outcome is significantly different from 28.26 cm squared.
B) 31.79 cm squared
This option is correct. By substituting the radius of 4.5 cm into the area formula \( A = \pi (4.5)^2 \), the calculation results in approximately 31.79 cm squared, confirming this as the accurate area of the circle.
C) 15.90 cm squared
This choice is incorrect as it represents a miscalculation of the area. The area formula clearly shows that the correct area must be larger, as \( 4.5^2 \) multiplied by \( \pi \) does not yield 15.90 cm squared.
D) 63.59 cm squared
This option is also incorrect. While it might seem plausible if one incorrectly used the radius or misapplied the formula, the actual area calculation with the given radius leads to a much smaller value than 63.59 cm squared.
Conclusion
The correct answer, 31.79 cm squared, is derived directly from applying the formula for the area of a circle using the radius provided. All other options either result from incorrect calculations or misunderstandings of the formula, highlighting that option B is the only viable answer based on the given information.
Answer: A
The probability that there are no repeated digits in a three-digit PIN is 0.72.
To calculate the probability that a randomly selected three-digit PIN using the digits 0–9 has no repeated digits, we find the total number of possible combinations and the favorable outcomes. The total combinations for a three-digit PIN where digits can be repeated is 10^3 (1000). For no repetition, the first digit has 10 options, the second has 9, and the third has 8, giving us 10 * 9 * 8 = 720 favorable outcomes. Therefore, the probability is 720/1000, which simplifies to 0.72.
A) 0.72
This option is correct as it accurately reflects the probability of selecting a three-digit PIN with no repeated digits. The calculation shows that there are 720 valid combinations out of 1000 total combinations, leading to a probability of 0.72.
B) 0.7
Option B is incorrect because it underestimates the probability of selecting a three-digit PIN without repeated digits. The accurate calculation yields 0.72, not 0.7, making this option an inaccurate representation of the probability.
C) 0.3
This option is incorrect as it significantly misrepresents the probability of selecting a three-digit PIN with no repeated digits. The calculation indicates a much higher probability of 0.72, thus making this option not viable.
D) 1
Option D is incorrect because a probability of 1 would suggest that every randomly selected PIN must have no repeated digits, which is not the case. Given the total combinations, the probability is actually 0.72, indicating that while it is likely, it is not guaranteed.
Conclusion
The correct answer of 0.72 accurately represents the probability of selecting a three-digit PIN with no repeated digits, derived from the proper calculation of favorable outcomes over total outcomes. All other options fail to reflect the mathematical reality of the situation, either underestimating or overestimating the likelihood of no repeated digits in the PIN.
3. Which of the following pairs of angles are supplementary? (Choose all that apply.)
Answer: D
60 degrees and 120 degrees are supplementary angles.
Supplementary angles are defined as two angles whose measures add up to 180 degrees. In this case, the pair of angles 60 degrees and 120 degrees sum to 180 degrees, which confirms that they are indeed supplementary.
A) 45 degrees and 45 degrees
This pair adds up to 90 degrees (45 + 45), which does not satisfy the definition of supplementary angles. Therefore, these angles are not supplementary.
B) 72 degrees and 100 degrees
The sum of 72 degrees and 100 degrees is 172 degrees (72 + 100), which also does not equal 180 degrees. Hence, this pair is not supplementary.
C) 35 degrees and 55 degrees
The total for this pair is 90 degrees (35 + 55), failing to reach the 180 degrees required for supplementary angles. Thus, this option is incorrect.
D) 60 degrees and 120 degrees
This pair of angles sums to 180 degrees (60 + 120), confirming that they meet the criteria for supplementary angles. Therefore, this option is correct.
Conclusion
The only pair of angles that qualifies as supplementary in this question is 60 degrees and 120 degrees. The other options do not add up to 180 degrees and thus fail to meet the definition of supplementary angles. This reinforces the understanding of how angle pairs are classified based on their sums.
4. Given these two quadrilaterals, find the value of
Answer: A
The value of the given quadrilaterals is 12.
The correct answer is 12, which is derived from the calculations based on the properties and dimensions of the two quadrilaterals provided.
A) 12
Option A is correct as it accurately reflects the calculated value based on the properties of the two quadrilaterals. The dimensions or angles provided in the problem lead to the conclusion that 12 is the valid solution when applying relevant geometric formulas.
B) 16
Option B is incorrect. This value does not align with the calculations derived from the properties of the quadrilaterals. It may suggest a misunderstanding of the dimensions or relationships between the shapes involved in the problem.
C) 20
Option C is also incorrect. The calculation leading to this option may reflect an error in measurement or application of the geometric principles relevant to these quadrilaterals. Therefore, it does not represent the proper solution.
D) 22
Option D is not correct as well. Choosing this value indicates a significant deviation from the calculations based on the properties of the quadrilaterals, which suggests that the necessary relationships have not been correctly applied.
Conclusion
The value of 12 is the definitive answer as it is supported by the geometric properties of the two quadrilaterals in question. All other options fail to satisfy the conditions set forth in the problem, demonstrating a misunderstanding or miscalculation of the relationships within the shapes.
Answer: C
Triangle C creates an enlarged similar triangle with a scale factor of 2.
The triangle with vertices (2,4), (6,6), and (10,16) is an enlarged version of the original triangle with vertices (1,2), (3,4), and (5,8) with a scale factor of 2.
A) (1,1), (3,2), (5,4)
This triangle does not maintain similarity to the original triangle as the vertices do not correspond to a uniform scale factor from the original triangle. The distances and angles between the vertices do not match the proportionality required for similarity.
B) (1,4), (3,6), (5,10)
Although this triangle has some proportionality, it does not maintain the correct scale factor of 2 from the original triangle. The ratios of the sides do not correspond to the necessary enlargement.
C) (2,4), (6,6), (10,16)
This triangle is derived by multiplying each vertex of the original triangle by the scale factor of 2. Each vertex corresponds to the original triangle's vertices, confirming that the triangles are similar and enlarged by the correct factor.
D) (3,4), (5,6), (7,10)
This triangle fails to maintain the necessary properties of similarity to the original triangle. The vertices do not reflect a consistent scale factor, resulting in a triangle that is neither similar nor correctly enlarged.
Conclusion
Triangle C is the only option that accurately represents an enlarged similar triangle with a scale factor of 2. The other options either fail to maintain the necessary proportionality or do not correspond to the defined scale factor, thus confirming that option C is definitively the correct choice.
Answer: D
The area of the walkway around the pool is 240 ft squared.
To find the area of the walkway, we first calculate the total area including the pool and the walkway, and then we subtract the area of the pool itself. The dimensions of the pool with the walkway are 20 ft by 44 ft, resulting in a total area of 880 ft squared, and subtracting the pool area of 640 ft squared gives us 240 ft squared for the walkway.
A) 880 ft squared
This option represents the total area of the pool and the walkway combined, not just the walkway itself. Therefore, it is incorrect as it does not answer the question regarding the area of the walkway alone.
B) 116 ft squared
This value does not correspond to any calculated area related to the pool or walkway. It is significantly less than the area of the walkway and does not align with the dimensions provided, making it an incorrect choice.
C) 640 ft squared
This option reflects the area of the pool itself, which measures 16 ft by 40 ft. Since the question specifically asks for the area of the walkway, this option is incorrect as it does not account for the additional space around the pool.
D) 240 ft squared
This is the correct answer as it accurately represents the area of the walkway surrounding the pool. By calculating the total area including the walkway (20 ft by 44 ft = 880 ft squared) and subtracting the area of the pool (640 ft squared), we arrive at 240 ft squared for the walkway.
Conclusion
The area of the walkway is definitively calculated as 240 ft squared by correctly applying the formula for the area of rectangles and understanding the relationship between the dimensions of the pool and the walkway. All other options either represent the area of the pool, the total area, or an incorrect calculation, confirming that D is the only valid choice.
Answer: A
The length of the remaining piece is 0.6 m.
To find the length of the remaining piece of rope, we need to subtract the lengths of the three known pieces from the total length of the rope. The total length of the rope is 6 m, and the sum of the three pieces (1.2 m + 1.5 m + 2.7 m) is 5.4 m. Therefore, the remaining piece measures 0.6 m.
A) 0.6 m
This option is correct because when we add the lengths of the three pieces (1.2 m + 1.5 m + 2.7 m), we get 5.4 m. Subtracting this from the total length of the rope (6 m) gives us 0.6 m, which is the length of the remaining piece.
B) 6 m
This option is incorrect as it suggests that the remaining piece is equal to the total length of the rope. Since the rope is cut into four pieces, the remaining piece cannot be the same length as the original rope.
C) 60 m
This option is incorrect because it presents a length that is significantly greater than the total length of the rope. It is not possible for any piece to exceed the total length of the rope, which is 6 m.
D) 600 m
This option is also incorrect as it suggests an unrealistic length that far exceeds the original 6 m rope. No piece can be longer than the total length of the material available.
Conclusion
The correct answer is 0.6 m, as it accurately reflects the remaining length after accounting for the lengths of the other pieces. All other options misinterpret the problem by either suggesting lengths that are not feasible based on the total length of the rope or are simply incorrect calculations.
Answer: B
Mei is correct, because her table requires 100 in squared of material.
Mei's square table has an area of 100 square inches, calculated as side length squared (10 in x 10 in = 100 in²). This area indicates the amount of material required to cover her table.
A) Mei, because her table requires 40 in squared of material
This option is incorrect because the area of Mei's square table is not 40 square inches. The correct calculation for the area of a square is the side length squared, which in this case is 10 in x 10 in, equaling 100 square inches.
B) Mei, because her table requires 100 in squared of material
This option is correct. Mei's square table, with each side measuring 10 inches, has an area of 100 square inches (10 in x 10 in = 100 in²), making it clear that her table requires this amount of material to cover.
C) Omar, because his table requires 314.16 in squared of material
This option is incorrect. While Omar's circular table does have a larger area, the area calculated is not precise. The area of a circle is calculated using the formula A = πr². With a diameter of 10 inches, the radius is 5 inches, leading to an area of approximately 78.54 square inches, not 314.16.
D) Omar, because his table requires 31.79 in squared of material
This option is also incorrect. The area of Omar's circular table is not 31.79 square inches. Using the correct formula A = πr², the area comes out to approximately 78.54 square inches, which is still less than Mei's table area.
Conclusion
Mei is correct in her assertion that her square table requires more material, as it has an area of 100 square inches. In contrast, Omar's circular table, with an area of approximately 78.54 square inches, requires less material. Thus, Mei's calculations definitively show her table needing more covering material compared to Omar's.
9. How many lines of symmetry does this shape have?
Answer: C
This shape has 2 lines of symmetry.
The shape in question possesses exactly two lines of symmetry, which allows it to be divided into two identical halves along those lines.
A) 4
Option A is incorrect because the shape does not have four lines of symmetry. A figure with four lines of symmetry would typically be a square or a specific type of rectangle, which does not apply to the shape being analyzed.
B) 8
Option B is also incorrect, as a shape with eight lines of symmetry would be highly regular, such as an octagon. The current shape does not meet the criteria for having that many lines of symmetry.
C) 2
Option C is correct because the shape can be divided into two identical sections along two specific lines, confirming that it has two lines of symmetry. This characteristic is typical for certain types of symmetrical shapes, such as a rectangle or an isosceles triangle.
D) 1
Option D is incorrect since a single line of symmetry would imply that the shape can only be mirrored along one axis. However, the shape described here has two distinct lines of symmetry, not just one.
Conclusion
The correct answer is definitively C, as the shape has two lines of symmetry, allowing for a clear division into identical halves. Options A, B, and D fail because they misrepresent the number of symmetries present in the shape, highlighting the importance of accurately identifying symmetry in geometric figures.
Answer: C
The data set is skewed right.
Given that the mean (14.5) is greater than the median (11.2), this indicates that the data set has a longer tail on the right side, which is characteristic of a right-skewed distribution.
A) Bimodal
A bimodal distribution has two modes, or peaks, in its frequency distribution. Since the question provides only the mean and median without any information about the frequency of values, there is no evidence to suggest that the data set has two distinct peaks, making this option incorrect.
B) Skewed left
A left-skewed distribution would have a mean that is less than the median, indicating that the tail on the left side is longer. In this case, since the mean is greater than the median, this option is not applicable to the data set.
C) Skewed right
This is the correct option. When the mean is greater than the median, it suggests that there are higher values pulling the mean upwards, which is typical of a right-skewed distribution. This means that the bulk of the data is concentrated on the left side, with a few higher values extending the tail on the right.
D) Symmetrical
A symmetrical distribution would have the mean and median equal, indicating that the data is evenly distributed around the center. Since the mean and median in this data set are different, it cannot be classified as symmetrical.
Conclusion
The data set is definitively classified as skewed right due to the mean being higher than the median, which indicates a longer tail on the right. Options A, B, and D fail to accurately describe the distribution's characteristics, reinforcing that C is the only correct choice.