Linear Systems- Modeling with Systems
A lighthouse is located at (1, 2) in a coordinate system measured in miles. A sailboat starts at (-7, 8) and sails in a positive x-direction along a path that can be modeled by a quadratic function with a vertex at (2, -6). Which system of equations can be used to determine whether the boat comes within 5 miles of the lighthouse?
(x - 1)² + (y - 2)² = 5 y = 14/81(x - 2)² - 6 (x - 1)² + (y - 2)² = 25 y = 14/81(x - 2)² - 6 (x - 1)² + (y - 2)² = 25 y = -14/81(x + 7)² + 8
The length of a rectangular field is 20 less than its width. The area of the field is 12,000 ft². What is the width of the field?
120 ft
Hector needs 84 feet of fencing to enclose his rectangular garden. The length of his garden is 12 feet less than twice its width. What is the width of his garden?
18 ft
A rectangular swimming pool has a perimeter of 96 ft. The area of the pool is 504 ft2. Which system of equations models this situation correctly?
2l + 2w = 98 lw = 504
At a skills competition, a target is being lifted into the air by a cable at a constant speed. An archer standing on the ground launches an arrow toward the target. The system of equations below models the height, in feet, of the target and the arrow t seconds after it was fired. Which statement most likely describes the situation modeled by this system?
8 32
A basketball player is shooting a basketball toward the net. The height, in feet, of the ball t seconds after the shot is modeled by the equation h = 6 + 30t - 16t². Two-tenths of a second after the shot is launched, an opposing player leaps up to block the shot. The height of the shot blocker's outstretched hands t seconds after he leaps is modeled by the equation h = 9 + 25t - 16t². If the ball reaches the net 1.7 seconds after the shooter launches it, does the leaping player block the shot?
No, the shot is not blocked.
The first equation in the system models the heights in feet, h, of a falling baseball as a function of time, t. The second equation models the heights in feet, h, of the glove of a player leaping up to catch the ball as a function of time, t. Which statement describes the situation modeled by this system?
The height of the baseball is 35 feet at the moment the player begins to leap.
The center of an ice rink is located at (0, 0) on a coordinate system measured in meters. Susan is skating along a path that can be modeled by the equation y = 6x - x² - 5. Luke starts at (10, -21) and skates along a path that can be modeled by a quadratic function with a vertex at (8, -9). If the rink is a circle with a radius of 35 meters, which statement best interprets the solution(s) of a system of equations modeling the paths of the skaters?
The skaters path intersect twice, but only one of those points is inside the rink.
The first equation in the system models the heights, h, of a falling volleyball as a function of time, t. The second equation models the heights, h, of the hands of a player jumping up to spike the ball as a function of time, t. Which statement describes the situation modeled by this system?
The volleyball is 14 feet above the ground at the instant the player begins her jump.
In the system of equations below, y represents the income from selling and the cost to produce x sweatshirts with team logos on them for a professional sports league. What does the solution of the system represent in this context?
_the number of sweatshirts that generate the maximum income the number of sweatshirts that generate the minimum cost the number of sweatshirts for which cost and income are equal
The local community theater sold a total of 240 tickets for Saturday night's performance. They sold 180 more full-price tickets than discount tickets. Which system of equations can be used to model this situation?
f + d = 240 f - d = 180
Graham and Hunter are circus performers. A cable lifts Graham into the air at a constant speed of 1.5 ft/s. When Graham's arms are 18 ft above the ground, Hunter, who is standing directly underneath Graham, throws Graham a ball as the cable continues to lift him higher. Hunter throws the ball from a position 5 ft above the ground with an initial velocity of 24 ft/s. Which system of equations can be used to model this situation?
h = 18 + 1.5t h = 5 + 24t - 16t²
A radio tower is located on a coordinate system measured in miles. The range of a signal in a particular direction is modeled by a quadratic function where the boundary of the signal starts at the vertex at (4, 2). It passes through the point (5, 4). A linear road connects points (-3, 7) and (8, 2). Which system of equations can be used to determine whether the road intersects the boundary of the tower's signal?
y - 2(x - 4)² = 2 5x + 11y = 62
Two boats depart from a port located at (-8, 1) in a coordinate system measured in kilometers and travel in a positive x-direction. The first boat follows a path that can be modeled by a quadratic function with a vertex at (1, 10), whereas the second boat follows a path that can be modeled by a quadratic function with a vertex at (0, -7). Which system of equations can be used to determine whether the paths of the boats cross?
y = -¹/₉(x - 1)² + 10 y = ¹/₈x² - 7
A company plans to sell a new type of vacuum cleaner for $280 each. The company's financial planner estimates that the cost, y, of manufacturing the vacuum cleaners is a quadratic function with a y-intercept of 11,000 and a vertex of (500, 24,000). Which system of equations can be used to determine how many vacuums must be sold for the company to make a profit?
y = 280x y = -0.052(x-500)² + 24,000
A coordinate grid is mapped on a video game screen, with the origin in the lower-left corner. A game designer programs a helicopter to follow a path that can be modeled by a quadratic function with a vertex at (16, 20) and passing through the point (4, 25). She also programs an airplane to move along a linear path that passes through the points (0, 18) and (30, 20). Which system of equations can be used to determine whether the paths of the helicopter and airplane cross?
y = ¹/₁₅x + 18 y = ⁵/₁₄₄(x - 16)² + 20