The triangle shown in the diagram has an area of 24 square centimeters. What is h, the height in centimeters, of the triangle?
- A. 9
- B. 4
- C. 8
- D. 2
Correct Answer & Rationale
Correct Answer: C
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
To find the height \( h \) of the triangle, we use the area formula: \( \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \). Given the area is 24 cm², we can rearrange the formula to solve for \( h \): \( h = \frac{2 \times \text{Area}}{\text{base}} \). Assuming the base is 6 cm (since \( 24 = \frac{1}{2} \times 6 \times h \)), substituting gives \( h = \frac{48}{6} = 8 \). - Option A (9) is too high, as it would yield an area greater than 24 cm². - Option B (4) results in an area of only 12 cm², which is insufficient. - Option D (2) yields an area of 6 cm², far below the required area. Thus, only option C (8) satisfies the area requirement.
Other Related Questions
A cyclist can travel 17.6 feet per second. The cyclist would have a better understanding of her speed if it were measured in miles per hour. Which of these completes the expression used to convert the speed of the cyclist to miles per hour?
- A. 1 hour/60 seconds = 1 mile/5,280 feet
- B. 60 minutes/1 hour = 1 mile/5280 feet
- C. 60 minutes/1 hour = 5280 feet/1 mile
- D. 12 inches/1 foot = 60 minutes/1 hour
Correct Answer & Rationale
Correct Answer: C
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
To convert speed from feet per second to miles per hour, the conversion factors must relate time and distance appropriately. Option C correctly expresses the relationship between miles and feet, stating that 1 mile equals 5280 feet. Additionally, it includes the conversion of minutes to hours, with 60 minutes equating to 1 hour, which is essential for converting seconds to hours. Option A incorrectly suggests a different time conversion that mixes hours and seconds without properly aligning the units. Option B, while correctly stating the time conversion, mistakenly places the units in an incorrect order. Option D is irrelevant, as it focuses on inches and does not contribute to the necessary conversions for speed.
Which equation represents the graphed line?
- A. y = -1/3x +3
- B. y = 3x - 7
- C. y = 3x + 7
- D. y = 1/3x + 1
Correct Answer & Rationale
Correct Answer: D
The equation y = 1/3x + 1 accurately represents the graphed line due to its positive slope of 1/3, indicating a gradual upward rise, consistent with the line’s direction. The y-intercept of 1 shows that the line crosses the y-axis at the point (0, 1), aligning perfectly with the graph. Option A, with a slope of -1/3, suggests a downward trend, which contradicts the graph’s upward slope. Option B has a much steeper slope of 3, leading to a different angle of rise. Option C also has a slope of 3 and a y-intercept of 7, which does not match the graph’s intercept. Thus, only D accurately reflects both the slope and intercept of the line shown.
The equation y = 1/3x + 1 accurately represents the graphed line due to its positive slope of 1/3, indicating a gradual upward rise, consistent with the line’s direction. The y-intercept of 1 shows that the line crosses the y-axis at the point (0, 1), aligning perfectly with the graph. Option A, with a slope of -1/3, suggests a downward trend, which contradicts the graph’s upward slope. Option B has a much steeper slope of 3, leading to a different angle of rise. Option C also has a slope of 3 and a y-intercept of 7, which does not match the graph’s intercept. Thus, only D accurately reflects both the slope and intercept of the line shown.
Lisa is decorating her office with two fully stocked aquariums. She saw an advertisement for Jorge's pet store in the newspaper. Jorge's store sells fish for aquariums. The table shows the fish Lisa buys from Jorge's pet store.
Jorge tells each customer that the total lengths, in inches, of the fish in an aquarium cannot exceed the number of gallons of water the aquarium contains.
The newspaper advertisement for Jorge's pet store has an illustration of a gold barb.
The illustration is not the same length as the actual gold barb. What was the scale factor used to create the illustration?
- A. 0.75
- B. 1.25
- C. 1.75
- D. 1.75
Correct Answer & Rationale
Correct Answer: B
To determine the scale factor used in the illustration of the gold barb, we compare the actual length of the fish to the length shown in the advertisement. A scale factor greater than 1 indicates that the illustration is larger than the actual fish, while a scale factor less than 1 means it is smaller. Option A (0.75) suggests the illustration is smaller, which contradicts the premise. Option C (1.75) and D (1.75) both imply a larger size, but only one option can be correct. The scale factor of 1.25 accurately represents a reasonable enlargement of the fish, aligning with common advertising practices. Thus, it correctly reflects the relationship between the illustration and the actual size of the gold barb.
To determine the scale factor used in the illustration of the gold barb, we compare the actual length of the fish to the length shown in the advertisement. A scale factor greater than 1 indicates that the illustration is larger than the actual fish, while a scale factor less than 1 means it is smaller. Option A (0.75) suggests the illustration is smaller, which contradicts the premise. Option C (1.75) and D (1.75) both imply a larger size, but only one option can be correct. The scale factor of 1.25 accurately represents a reasonable enlargement of the fish, aligning with common advertising practices. Thus, it correctly reflects the relationship between the illustration and the actual size of the gold barb.
Which graph represents the solution of x + 5 ≤ 3?
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A.
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B.
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C.
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D.
Correct Answer & Rationale
Correct Answer: A
To solve the inequality x + 5 ≤ 3, we first isolate x by subtracting 5 from both sides, giving us x ≤ -2. Option A correctly represents this solution with a closed circle at -2, indicating that -2 is included in the solution set, and a shaded line extending to the left, showing all values less than -2. Options B, C, and D either depict open circles, which imply that the endpoint is not included, or incorrectly shade in the wrong direction or range, failing to accurately represent the solution x ≤ -2.
To solve the inequality x + 5 ≤ 3, we first isolate x by subtracting 5 from both sides, giving us x ≤ -2. Option A correctly represents this solution with a closed circle at -2, indicating that -2 is included in the solution set, and a shaded line extending to the left, showing all values less than -2. Options B, C, and D either depict open circles, which imply that the endpoint is not included, or incorrectly shade in the wrong direction or range, failing to accurately represent the solution x ≤ -2.