Exploring Quadratic Inequalities: Graphing & Solving Challenges

A farmer wants to create a rectangular garden with an area of at least 200 square meters. If the length of the garden is represented by x meters, write a quadratic inequality to represent the possible values for the width, w, in terms of x. Graph the inequality.

• w < 200/x

• w = 200/x

• w <= 200/x

• w >= 200/x

A company produces a certain product and finds that their profit, P, can be modeled by the inequality P(x) = -2x^2 + 8x - 10, where x is the number of units sold. Determine the range of units that must be sold to ensure a profit greater than zero. Solve the inequality and graph the solution.

• Selling 3 units guarantees a profit of $5.

• No units can be sold to ensure a profit greater than zero.

• A profit greater than zero can be achieved by selling 5 units.

• The company will break even at 10 units sold.

A basketball player can jump to a height modeled by the equation h(t) = -4.9t^2 + 10t, where h is the height in meters and t is the time in seconds. Determine the time intervals during which the player is above 5 meters. Solve the quadratic inequality and graph the results.

• The player is above 5 meters during the time intervals (0, 2) seconds.

• The player is above 5 meters during the time intervals (0.5, 4.08) seconds.

• The player is above 5 meters during the time intervals (2, 5) seconds.

• The player is above 5 meters during the time intervals (1, 3) seconds.

A car's speed, s, is affected by the weight of the cargo, w, according to the inequality s(w) = -0.5w^2 + 5w - 10. Find the range of weights that will allow the car to maintain a speed greater than 0. Solve the inequality and graph the solution set.

• (5, 8)

• (10, 15)

• (0, 2)

• (2, 10)

A rectangular swimming pool has a length that is 3 meters longer than its width. If the area of the pool must be less than 100 square meters, write a quadratic inequality to represent this situation. Solve the inequality and graph the solution.

• w > 15

• w = 10

• w < 5

• w < 10, where w is the width of the pool.

A projectile is launched from a height of 50 meters and its height, h, in meters after t seconds is given by h(t) = -5t^2 + 20t + 50. Determine the time intervals when the projectile is above 30 meters. Solve the quadratic inequality and graph the results.

• (3, 7)

• (2, 6)

• (1, 4)

• (0.17, 5.83)

A local theater has a seating capacity modeled by the inequality x^2 - 10x + 24 < 0, where x represents the number of rows. Determine the number of rows that can be added while keeping the seating capacity below the maximum. Solve the inequality and graph the solution.

• 4

• 1

• 3

• 2

A rectangular plot of land has a length that is twice its width. If the area of the plot must be at least 500 square meters, write a quadratic inequality to represent the possible dimensions. Solve the inequality and graph the solution set.

• w < 5√10, length = 2w < 10√10

• w = 5, length = 10

• w >= 5√10, length = 2w >= 10√10

• w <= 0, length = 0

A toy company finds that the number of toys sold, n, can be modeled by the inequality n = -3x^2 + 12x + 5, where x is the price of the toy. Determine the price range that will result in selling more than 0 toys. Solve the inequality and graph the results.

• (0.5, 3)

• (1, 2)

• (2, 4)

• (3, 5)

A water tank can hold a maximum volume of 500 liters. The volume, V, in liters, can be modeled by the inequality V = -2x^2 + 20x, where x is the height of the water in meters. Find the height range that keeps the volume below 500 liters. Solve the inequality and graph the solution.

• Height range: 0 < x < 10 meters

• Height range: x < 5 meters

• Height range: x = 15 meters

• Height range: x > 10 meters