Mastering Quadratic Functions: Maxima and Area Problems

A ball is thrown upwards from a height of 2 meters with an initial velocity of 10 m/s. The height of the ball after t seconds is given by the equation h(t) = -5t^2 + 10t + 2. What is the maximum height the ball reaches?

• 10 meters

• 7 meters

• 3 meters

• 5 meters

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is 40 square meters, what are the dimensions of the garden?

• Width: 3 meters, Length: 6 meters

• Width: 5 meters, Length: 8 meters

• Width: 4 meters, Length: 7 meters

• Width: 6 meters, Length: 9 meters

The path of a projectile can be modeled by the equation h(t) = -4.9t^2 + 20t + 5, where h is the height in meters and t is the time in seconds. How long will it take for the projectile to hit the ground?

• 6.75 seconds

• 4.56 seconds

• 5.10 seconds

• 3.25 seconds

A company finds that the profit P (in dollars) from selling x items is given by the equation P(x) = -2x^2 + 40x - 50. How many items should the company sell to maximize its profit?

• 5

• 20

• 10

• 15

A car's distance from a starting point can be modeled by the equation d(t) = -3t^2 + 12t, where d is the distance in meters and t is the time in seconds. At what time does the car reach its maximum distance?

• 4 seconds

• 3 seconds

• 1 second

• 2 seconds

The height of a water fountain can be modeled by the equation h(t) = -16t^2 + 32t + 5. What is the time at which the fountain reaches its maximum height?

• 1

• 0

• 2

• 3

A farmer wants to create a rectangular pen with a fixed perimeter of 100 meters. What dimensions will maximize the area of the pen?

• 15 meters by 35 meters

• 10 meters by 40 meters

• 25 meters by 25 meters

• 20 meters by 30 meters

The area A of a triangle can be expressed as A = 1/2 * base * height. If the height is modeled by the equation h(b) = -b^2 + 10b, where b is the base length, what is the maximum area of the triangle?

• 15.00

• 40.00

• 33.33

• 25.00

A toy rocket is launched and its height in meters is given by the equation h(t) = -4.9t^2 + 30t + 1. How long will the rocket be in the air before it hits the ground?

• 7.50 seconds

• 5.00 seconds

• 8.10 seconds

• 6.24 seconds

A rectangular swimming pool has a length that is twice its width. If the area of the pool is 200 square meters, what are the dimensions of the pool?

• Width: 10 meters, Length: 20 meters

• Width: 5 meters, Length: 10 meters

• Width: 15 meters, Length: 30 meters

• Width: 8 meters, Length: 16 meters