Modeling with Systems Quiz

Study Notes

Boats Departing from Port

Two boats leave from coordinates (-8, 1) traveling positively along the x-axis.
First boat's path modeled by a quadratic function: ( y = -\frac{1}{9}(x - 1)^2 + 10 ) (vertex at (1, 10)).
Second boat's path modeled by a quadratic function: ( y = \frac{1}{8}x^2 - 7 ) (vertex at (0, -7)).
System of equations used to determine intersection of the paths.

Archery Competition

In a competition, a target is raised at a constant speed.
An archer fires an arrow toward the target.
Arrow's situation modeled by an equation indicating an initial upward velocity of 32 ft/s.

Baseball Catching Scenario

First equation captures height of a falling baseball as a function of time.
Second equation represents height of a glove of a player jumping to catch the ball.
Key moment: the baseball is 35 feet high when the player starts to leap.

Vacuum Cleaner Profit Analysis

A new vacuum cleaner priced at $280 each.
Cost function is quadratic with a y-intercept of 11,000 and a vertex at (500, 24,000).
System of equations for determining sales needed for profitability:
- ( y = 280x )
- ( y = -0.052(x-500)^2 + 24,000 )

Rectangular Swimming Pool

Swimming pool with a perimeter of 96 ft and area of 504 ft².
Equation setup:
- ( 2l + 2w = 98 )
- ( lw = 504 )

Community Theater Ticket Sales

Total of 240 tickets sold for a performance.
The relationship between ticket types: 180 more full-price tickets than discount tickets.
Modeling equations include:
- ( f + d = 240 )
- ( f - d = 180 )

Fencing for Hector's Garden

Hector requires 84 feet of fencing for a rectangular garden.
Length is 12 feet less than twice the width.
Result: Width of the garden is calculated at 18 ft.

Lighthouse and Sailboat Interaction

Lighthouse located at (1, 2) miles; boat starts at (-7, 8) miles and travels positively on x-axis.
Boat's path modeled by ( y = \frac{14}{81}(x - 2)^2 - 6 ) with vertex at (2, -6).
Determine the boat's proximity to the lighthouse through the equation ( (x-1)^2 + (y-2)^2 = 25 ) (5-mile radius).

Rectangular Field Dimensions

Length of a rectangular field is 20 feet less than its width.
Area of the field is given as 12,000 ft².
Width is determined to be 120 ft.

Radio Tower Signal Coverage

Radio tower's signal range modeled by a quadratic function, vertex at (4, 2) and passes through (5, 4).
Linear road defined by points (-3, 7) and (8, 2).
To check if the road intersects the signal boundary, use the equations:
- ( y - 2 = (x - 4)^2 )
- ( 5x + 11y = 62 )

Description

Test your knowledge with this quiz on modeling systems involving quadratic functions and coordinate systems. Determine the system of equations needed to analyze the paths of two boats departing from a port, focusing on their intersection. Perfect for students in mathematics or physics classes.