Solving Real World Inequalities in Quadratic Contexts

A farmer wants to create a rectangular garden with an area of at least 200 square meters. If the length of the garden is 5 meters more than its width, write an inequality to represent the relationship between the length and width of the garden.

• w * (w + 5) >= 200

• w * (w - 5) >= 200

• w * (w + 10) >= 200

• w + (w + 5) <= 200

A ball is thrown into the air, and its height in meters can be modeled by the equation h(t) = -4.9t^2 + 20t + 1, where t is the time in seconds. Determine the maximum height the ball reaches and write an inequality to find when the ball is above 10 meters.

• Maximum height: 41.04 meters; Inequality: -4.9t^2 + 20t - 9 > 0

• Maximum height: 30.5 meters; Inequality: -4.9t^2 + 20t - 5 > 0

• Maximum height: 25.0 meters; Inequality: -4.9t^2 + 20t - 15 > 0

• Maximum height: 35.2 meters; Inequality: -4.9t^2 + 20t - 20 > 0

A company produces a certain product, and the profit in dollars can be modeled by the equation P(x) = -2x^2 + 40x - 50, where x is the number of units sold. Write an inequality to find the number of units that must be sold to achieve a profit greater than $0.

• -2x^2 + 40x + 50 < 0

• 2x^2 + 40x - 50 > 0

• 2x^2 - 40x + 50 < 0

• 2x^2 - 40x - 50 > 0

A car's value decreases over time and can be modeled by the inequality V(t) = -3t^2 + 30t + 5000, where V is the value in dollars and t is the age of the car in years. Determine the age of the car when its value is at least $4000.

• The car is at least $4000 in value for ages up to 20 years.

• The car is at least $4000 in value for ages up to 10 years.

• The car is at least $4000 in value for ages up to approximately 36.78 years.

• The car is at least $4000 in value for ages up to 50 years.

A swimming pool can hold a maximum of 5000 liters of water. If the pool is being filled at a rate of 200 liters per hour, write an inequality to determine how many hours it will take to fill the pool to at least 3000 liters.

• h < 15

• h > 15

• h <= 15

• h >= 15

A projectile is launched from the ground and its height can be modeled by the equation h(t) = -5t^2 + 20t, where t is the time in seconds. Write an inequality to find the time when the projectile is at least 15 meters high.

• 5t^2 - 20t + 15 >= 0

• -5t^2 + 20t < 15

• 5t^2 - 20t - 15 <= 0

• 5t^2 - 20t + 15 <= 0

A rectangular field has a perimeter of 100 meters. If the length is represented as x and the width as y, write an inequality to express the area of the field being greater than 400 square meters.

• x(50 + x) > 400

• 2x + 2y < 100

• x(50 - x) > 400

• xy < 400

A local theater has a seating capacity of 300. If ticket sales can be modeled by the inequality 5x + 10y ≤ 300, where x is the number of adult tickets and y is the number of child tickets, determine the combinations of tickets that can be sold without exceeding capacity.

• (30, 20)

• (50, 10)

• (100, 0)

• Valid combinations of tickets (x, y) are: (60, 0), (58, 1), (56, 2), ..., (0, 30).

A water tank can hold a maximum of 1500 liters. If the tank is currently holding 600 liters and is being filled at a rate of 50 liters per hour, write an inequality to find out how long it will take to fill the tank to at least 1200 liters.

• t = 12

• t < 12

• t ≥ 12

• t ≤ 12

A rectangular plot of land has a length that is 3 meters longer than its width. If the area of the plot must be at least 150 square meters, write an inequality to represent the relationship between the length and width of the plot.

• w(w - 3) ≥ 150

• w(w + 3) ≥ 150

• w(w + 2) ≤ 150

• w(w + 5) ≥ 150