Modeling with Systems Quiz

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 = 25 y = 14/81 (x-2)^2 - 6

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?

The arrow is fired with an initial upward velocity of 32 ft/s

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 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

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