Maximizing Heights: Vertex and Quadratic Formula Challenges

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 vertex of this parabola, and what does it represent in this context?

• The vertex of the parabola is (0, 2), representing the initial height of the ball.

• The vertex of the parabola is (2, 12), representing the height at 2 seconds.

• The vertex of the parabola is (1, 7), representing the maximum height of the ball at 1 second.

• The vertex of the parabola is (1, 5), representing the height of the ball at 1 second.

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 using a quadratic equation. What is the width?

• 4 meters

• 8 meters

• 10 meters

• 6 meters

A company finds that the profit P (in dollars) from selling x items can be modeled by the equation P(x) = -2x^2 + 40x - 100. What is the maximum profit, and how many items must be sold to achieve it?

• Maximum profit is $100, achieved by selling 10 items.

• Maximum profit is $90, achieved by selling 8 items.

• Maximum profit is $120, achieved by selling 15 items.

• Maximum profit is $80, achieved by selling 5 items.

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. Determine the time at which the projectile reaches its maximum height and what that height is.

• The projectile reaches its maximum height at 3.0 seconds, with a height of 25.0 meters.

• The projectile reaches its maximum height at 2.04 seconds, with a height of approximately 21.04 meters.

• The projectile reaches its maximum height at 1.5 seconds, with a height of 18.5 meters.

• The projectile reaches its maximum height at 2.5 seconds, with a height of 19.0 meters.

A farmer wants to create a rectangular pen with an area of 120 square meters. If the length of the pen is 5 meters more than its width, find the dimensions of the pen. What is the width?

• 12 meters

• 8 meters

• 15 meters

• 10 meters

A water fountain is designed to have a parabolic shape, described by the equation y = -x^2 + 4x + 1. Identify the vertex of the parabola and explain its significance in the context of the fountain's height.

• The vertex of the parabola is (0, 1), showing the starting point of the fountain's height.

• The vertex of the parabola is (2, 5), representing the maximum height of the fountain.

• The vertex of the parabola is (1, 6), indicating the fountain's minimum height.

• The vertex of the parabola is (3, 4), which represents the average height of the fountain.

A car's profit can be modeled by the equation P(x) = -3x^2 + 30x - 45, where x is the number of cars sold. Use the quadratic formula to find the number of cars sold that maximizes profit.

• 5

• 10

• 7

• 3

The height of a bridge is modeled by the equation h(x) = -x^2 + 6x + 8, where x is the distance from the base of the bridge. Find the vertex of this parabola and interpret its meaning in the context of the bridge's height.

• The vertex of the parabola is (4, 12), indicating the maximum height of the bridge is 12 units at a distance of 4 units from the base.

• The vertex of the parabola is (5, 5), indicating the maximum height of the bridge is 5 units at a distance of 5 units from the base.

• The vertex of the parabola is (3, 17), indicating the maximum height of the bridge is 17 units at a distance of 3 units from the base.

• The vertex of the parabola is (2, 10), indicating the maximum height of the bridge is 10 units at a distance of 2 units from the base.

A toy rocket is launched, and its height in meters after t seconds is given by the equation h(t) = -4.9t^2 + 20t + 5. Calculate the time at which the rocket reaches its maximum height and the maximum height itself.

• Time: 1.5 seconds, Maximum Height: 22 meters

• Time: 2 seconds, Maximum Height: 20 meters

• Time: 2.04 seconds, Maximum Height: 25.4 meters

• Time: 3 seconds, Maximum Height: 30 meters

A rectangular swimming pool has a length that is 4 meters longer than its width. If the area of the pool is 96 square meters, find the dimensions of the pool using a quadratic equation. What is the width?

• 12 meters

• 8 meters

• 6 meters

• 10 meters