Modeling with Systems Answers

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 was on a path to reach the net 1.7 seconds after the shooter launches it, does the leaping player block the shot?

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?

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?

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?

The width of a rectangular field is 20 feet less than its length. The area of the field is 12,000 ft2. What is the length of the field?

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?

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?

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?

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