What is the area, in square inches, of a circle with diameter 2 inches?
- A. 6.28
- B. 3.14
- C. 1
- D. 12.56
Correct Answer & Rationale
Correct Answer: B
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. Given a diameter of 2 inches, the radius is 1 inch. Substituting this into the formula yields \( A = \pi (1)^2 = \pi \), which approximates to 3.14. Option A (6.28) incorrectly doubles the area, possibly confusing it with the circumference. Option C (1) neglects the use of \(\pi\), leading to an inaccurate calculation. Option D (12.56) mistakenly uses the formula for circumference, multiplying the diameter by \(\pi\) instead of squaring the radius. Thus, 3.14 accurately represents the area of the circle.
To find the area of a circle, the formula \( A = \pi r^2 \) is used, where \( r \) is the radius. Given a diameter of 2 inches, the radius is 1 inch. Substituting this into the formula yields \( A = \pi (1)^2 = \pi \), which approximates to 3.14. Option A (6.28) incorrectly doubles the area, possibly confusing it with the circumference. Option C (1) neglects the use of \(\pi\), leading to an inaccurate calculation. Option D (12.56) mistakenly uses the formula for circumference, multiplying the diameter by \(\pi\) instead of squaring the radius. Thus, 3.14 accurately represents the area of the circle.
Other Related Questions
Which graph shows a line described by 4x - 3y = 12?
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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: D
To determine which graph represents the line described by the equation 4x - 3y = 12, we can rearrange it into slope-intercept form (y = mx + b). This yields y = (4/3)x - 4. The slope (m) is 4/3, indicating the line rises 4 units for every 3 units it runs to the right, and the y-intercept (b) is -4, meaning the line crosses the y-axis at (0, -4). Option D correctly displays a line with a positive slope and a y-intercept at -4. Options A, B, and C either have the wrong slope or intercept, indicating they do not accurately represent the given equation.
To determine which graph represents the line described by the equation 4x - 3y = 12, we can rearrange it into slope-intercept form (y = mx + b). This yields y = (4/3)x - 4. The slope (m) is 4/3, indicating the line rises 4 units for every 3 units it runs to the right, and the y-intercept (b) is -4, meaning the line crosses the y-axis at (0, -4). Option D correctly displays a line with a positive slope and a y-intercept at -4. Options A, B, and C either have the wrong slope or intercept, indicating they do not accurately represent the given equation.
A bag of dog food weighs 40 pounds. The amount of food in the bag is more than 3 times the amount needed to feed a dog for one week. Which inequality can be used to determine the possible values for p, the pounds of food needed to feed the dog for one week?
- A. p < 3(40)
- B. 3p < 40
- C. p > 3(40)
- D. 3p > 40
Correct Answer & Rationale
Correct Answer: D
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
To find the amount of food needed for one week, we know that the total weight of the dog food (40 pounds) is more than three times the weekly requirement (3p). Therefore, the relationship can be expressed as 3p < 40, indicating that the total food exceeds three times the weekly amount. Option A (p < 3(40)) incorrectly suggests that the weekly requirement is less than three times the total weight, which is not supported by the problem statement. Option B (3p < 40) misrepresents the relationship, as it implies the total food is less than three times the weekly need, contradicting the given information. Option C (p > 3(40)) inaccurately states that the weekly requirement exceeds three times the total weight, which is impossible given the context. Thus, the correct inequality is 3p > 40, indicating the total food is indeed more than three times the weekly requirement.
Solve the equation for x: ½ x + 9 = -2/3 x
- A. x=-9/7
- B. x=-54/7
- C. x=-6
- D. x=-54
Correct Answer & Rationale
Correct Answer: B
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).
To solve the equation \( \frac{1}{2}x + 9 = -\frac{2}{3}x \), start by eliminating the fractions. Multiply the entire equation by 6 (the least common multiple of 2 and 3) to obtain \( 3x + 54 = -4x \). Next, combine like terms: adding \( 4x \) to both sides gives \( 7x + 54 = 0 \), leading to \( 7x = -54 \) and thus \( x = -\frac{54}{7} \). Option A is incorrect as it simplifies to a different value. Option C, \( x = -6 \), does not satisfy the original equation. Option D, \( x = -54 \), is also incorrect as it does not balance the equation. Therefore, the only viable solution is \( x = -\frac{54}{7} \).
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.