Maximizing Area & Profit: Quadratics in Action

A ball is thrown upwards from a height of 2 meters with an initial velocity of 10 m/s. The height of the ball in meters after t seconds is given by the equation h(t) = -5t^2 + 10t + 2. How long will it take for the ball to hit the ground?

• 4.20 seconds

• 2.50 seconds

• 3.74 seconds

• 5.00 seconds

The area of a rectangular garden is represented by the equation A = x(10 - x), where A is the area in square meters and x is the width in meters. What are the dimensions of the garden that maximize the area?

• 4 meters by 6 meters

• 3 meters by 7 meters

• 2 meters by 8 meters

• 5 meters by 5 meters

A company finds that the profit P (in dollars) from selling x units of a product can be modeled by the equation P(x) = -2x^2 + 40x - 100. How many units should the company sell to maximize its profit?

• 20

• 10

• 5

• 15

The path of a projectile is modeled by the equation y = -4.9t^2 + 20t + 5, where y is the height in meters and t is the time in seconds. At what time does the projectile reach its maximum height?

• 2.04 seconds

• 3.00 seconds

• 4.10 seconds

• 1.50 seconds

A rectangular swimming pool has a length that is 3 meters longer than its width. If the area of the pool is 70 square meters, write a quadratic equation to find the dimensions of the pool. What are the dimensions?

• Width: 5 meters, Length: 8 meters

• Width: 6 meters, Length: 9 meters

• Width: 7 meters, Length: 10 meters

• Width: 8 meters, Length: 11 meters

The height of a parabolic arch is modeled by the equation h(x) = -x^2 + 6x + 8, where h is the height in meters and x is the horizontal distance in meters. What is the maximum height of the arch?

• 15 meters

• 17 meters

• 10 meters

• 20 meters

A farmer wants to create a rectangular field with a fixed perimeter of 100 meters. Express the area A of the field as a quadratic function of one side length x. What dimensions will maximize the area?

• The dimensions that maximize the area are 25 meters by 25 meters.

• 10 meters by 40 meters

• 15 meters by 35 meters

• 20 meters by 30 meters

The revenue R (in dollars) from selling x items is given by the equation R(x) = 50x - 0.5x^2. Determine the number of items that must be sold to maximize revenue and find the maximum revenue.

• 50 items, maximum revenue is $1250

• 30 items, maximum revenue is $600

• 70 items, maximum revenue is $1400

• 40 items, maximum revenue is $1000

A toy rocket is launched from a height of 1 meter with an initial velocity of 15 m/s. The height of the rocket after t seconds is given by the equation h(t) = -4.9t^2 + 15t + 1. How high will the rocket go?

• 10.5 meters

• 15 meters

• 8.75 meters

• 12.25 meters

A rectangular garden has a length that is twice its width. If the area of the garden is 200 square meters, write a quadratic equation to find the dimensions of the garden. What are the dimensions?

• Width: 10 meters, Length: 20 meters

• Width: 15 meters, Length: 30 meters

• Width: 5 meters, Length: 10 meters

• Width: 8 meters, Length: 16 meters