A diver jumps from a platform. The height, h meters, the diver is above the water t seconds after jumping is represented by h = -16t^2 + 16t + 6.5. To the near hundredth of a second, how many seconds after jumping is the diver 2.5 meters above the water?
- A. 2.79
- B. 1.32
- C. 2.83
- D. 1.21
Correct Answer & Rationale
Correct Answer: D
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
To find when the diver is 2.5 meters above the water, substitute h = 2.5 into the equation: \[ 2.5 = -16t^2 + 16t + 6.5. \] Rearranging gives: \[ -16t^2 + 16t + 4 = 0. \] Using the quadratic formula, we solve for t, yielding two potential solutions. The option D (1.21 seconds) is valid as it falls within the realistic time frame of the jump. Options A (2.79) and C (2.83) exceed the expected time of descent, while B (1.32) does not satisfy the equation, confirming that only D accurately represents the diver's position at 2.5 meters above the water.
Other Related Questions
An advertisement poster in the window of a shoe store is in the shape of a rectangle. The length of the poster is 9 less than 4 times the width. Which expression represents the length of the poster when w is the width
- A. 4w - 9
- B. 9 - 4w
- C. 4w + 9
- D. 9w - 4
Correct Answer & Rationale
Correct Answer: A
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
The expression for the length of the poster is determined by the relationship given in the problem. The length is described as "9 less than 4 times the width," which translates mathematically to \(4w - 9\). Option A (4w - 9) accurately reflects this relationship. Option B (9 - 4w) incorrectly suggests that the length is greater than 9 and decreases as width increases, which contradicts the problem's description. Option C (4w + 9) implies that the length increases by 9, rather than decreasing, which is not aligned with the original statement. Option D (9w - 4) introduces an incorrect multiplication factor and does not adhere to the given relationship, making it invalid.
A carpenter is installing shelves in 2 offices. Each office will have 4 shelves. The wood the carpenter wants to use comes in 6-foot-long boards. Each shelf is 2 ¼ feet long and is constructed from a single board. How many boards does the carpenter need to buy to make the shelves?
- A. 2
- B. 8
- C. 3
- D. 4
Correct Answer & Rationale
Correct Answer: D
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
To determine how many boards are needed, first calculate the total length of wood required for the shelves. Each office has 4 shelves, and with 2 offices, that totals 8 shelves. Each shelf is 2 ¼ feet long, which equals 2.25 feet. Therefore, the total length required is 8 shelves x 2.25 feet = 18 feet. Each board is 6 feet long. Dividing the total length (18 feet) by the length of each board (6 feet) gives 3 boards. However, since each board can only be used for one shelf, and we can't cut a board to make multiple shelves, we need to round up to the nearest whole number of boards needed, which is 4. - Option A (2 boards) is insufficient for the total length required. - Option B (8 boards) exceeds the necessary amount. - Option C (3 boards) miscalculates the total need based on the cut requirement. Thus, 4 boards are necessary to accommodate all shelves without waste.
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.
Type your answer in the box. You may use numbers, a decimal point (.), and/or negative sign (-) in your answer.
One evening on her walk, Laura walks across the field from point W back to the gate at point G. What is the distance she walks, in feet, from point W to point G?
Correct Answer & Rationale
Correct Answer: 250
To determine the distance Laura walks from point W to gate G, we can use the Pythagorean theorem. The field is a rectangle, and point W is at one corner. The length from W to G is half the length of the field (150 ft) and the width of the field (200 ft). Calculating the distance: Distance = √(150² + 200²) = √(22500 + 40000) = √62500 = 250 ft. Other options are incorrect because they do not accurately reflect the geometric relationship of the points. Distances such as 300 ft or 200 ft misinterpret the diagonal distance, while any number below 250 fails to account for both dimensions of the rectangle.
To determine the distance Laura walks from point W to gate G, we can use the Pythagorean theorem. The field is a rectangle, and point W is at one corner. The length from W to G is half the length of the field (150 ft) and the width of the field (200 ft). Calculating the distance: Distance = √(150² + 200²) = √(22500 + 40000) = √62500 = 250 ft. Other options are incorrect because they do not accurately reflect the geometric relationship of the points. Distances such as 300 ft or 200 ft misinterpret the diagonal distance, while any number below 250 fails to account for both dimensions of the rectangle.
Type your answer in the box. You may use numbers, a decimal point (-), and/or a negative sign (-) in your answer.
A truck driver sees a road sign warning of an 8% road incline. To the nearest tenth of a foot, what will be the change in the truck's vertical position, in feet, during the time it takes the truck's horizontal position to change by 1 mile? (1 mile = 5,280 ft)
Correct Answer & Rationale
Correct Answer: 422.4
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.
To determine the vertical change during a 1-mile horizontal distance on an 8% incline, we calculate the vertical rise using the formula: vertical rise = incline percentage × horizontal distance. Here, 8% as a decimal is 0.08, and the horizontal distance is 5,280 feet. Therefore, the vertical change is 0.08 × 5,280 = 422.4 feet. Other options are incorrect as they either miscalculate the incline percentage or the conversion of miles to feet. For instance, values significantly lower than 422.4 feet suggest a misunderstanding of the incline's impact, while options above this value imply an overestimation of the incline's effect on vertical change.