Modeling with Systems Answers

The first equation in the system models the height, h, of a falling volleyball as a function of time, t. The second equation models the height, 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?

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, where l is the length of the pool in feet and w is the width of the pool in feet?

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?

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?

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?

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?

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?

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?

In the first equation in the system of equations, y represents the money collected from selling sweatshirts. In the second equation, y represents the money spent 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?

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?