Exploring Quadratic Roots: Real-Life Applications and Insights

A ball is thrown upwards from a height of 5 meters. The height of the ball in meters after t seconds is given by the equation h(t) = -5t^2 + 20t + 5. Find the time when the ball hits the ground and interpret the significance of the roots.

• t = 4 seconds

• t = 2 seconds

• t = 6 seconds

• t = 8 seconds

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 54 square meters, find the dimensions of the garden by solving the quadratic equation formed. What do the roots represent in this context?

• Width: 5 meters, Length: 8 meters

• Width: 6 meters, Length: 9 meters

• Width: 7 meters, Length: 10 meters

• Width: 4 meters, Length: 7 meters

A company finds that the profit P (in thousands of dollars) from selling x units of a product can be modeled by the equation P(x) = -2x^2 + 40x - 150. Determine the number of units sold when the profit is zero and explain the significance of the roots.

• 5 and 15

• 10 and 20

• 3 and 12

• 0 and 30

The path of a projectile is modeled by the equation h(t) = -4.9t^2 + 20t + 1, where h is the height in meters and t is the time in seconds. Calculate the time when the projectile reaches the ground and discuss what this means in terms of the projectile's flight.

• Approximately 4.14 seconds

• Approximately 3.1 seconds

• Approximately 6.8 seconds

• Approximately 2.5 seconds

A car's distance from a starting point is modeled by the equation d(t) = 2t^2 - 16t + 30, where d is in meters and t is in seconds. Find the time when the car is at the starting point and analyze the roots in the context of the car's movement.

• The car is at the starting point at t = 0 seconds and t = 10 seconds.

• The car is at the starting point at t = 1 second and t = 6 seconds.

• The car is at the starting point at t = 3 seconds and t = 5 seconds.

• The car is at the starting point at t = 2 seconds and t = 4 seconds.

A swimming pool is being filled with water, and the volume V (in liters) can be modeled by the equation V(t) = -3t^2 + 30t + 10, where t is the time in minutes. Determine when the pool will be full and interpret the roots in relation to the filling process.

• The pool will be full at 20 minutes.

• The pool will be full at 15 minutes.

• The pool will be full at approximately 28.78 minutes.

• The pool will be full at 35 minutes.

The height of a projectile is modeled by the equation h(t) = -16t^2 + 64t + 80. Find the time when the projectile reaches its maximum height and when it hits the ground. Discuss the significance of these roots in the context of the projectile's motion.

• Maximum height at t = 2 seconds; hits the ground at t = 5 seconds.

• Maximum height at t = 3 seconds; hits the ground at t = 6 seconds.

• Maximum height at t = 2.5 seconds; hits the ground at t = 7 seconds.

• Maximum height at t = 1 second; hits the ground at t = 4 seconds.

A farmer has a rectangular field with a length that is 4 meters longer than its width. If the area of the field is 96 square meters, find the dimensions of the field and explain what the roots of the quadratic equation represent.

• Width: 8 meters, Length: 12 meters

• Width: 5 meters, Length: 9 meters

• Width: 10 meters, Length: 14 meters

• Width: 6 meters, Length: 10 meters

A toy rocket is launched, and its height in meters after t seconds is given by the equation h(t) = -5t^2 + 30t + 10. Calculate the time when the rocket lands and interpret the significance of the roots in terms of the rocket's flight.

• The rocket lands at 10 seconds.

• The rocket lands at 8.2 seconds.

• The rocket lands at 4.5 seconds.

• The rocket lands at approximately 6.77 seconds.

A rectangular pool has a length that is twice its width. If the area of the pool is 200 square meters, find the dimensions of the pool and analyze the roots of the quadratic equation formed in this context.

• Width: 10 meters, Length: 20 meters

• Width: 8 meters, Length: 16 meters

• Width: 15 meters, Length: 30 meters

• Width: 5 meters, Length: 10 meters