Factoring Quadratic Equations in Real-World Scenarios

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden can be expressed as (x)(x + 3) square meters, what are the dimensions of the garden?

• Width: x meters, Length: x + 3 meters

• Width: x + 3 meters, Length: x meters

• Width: x - 3 meters, Length: x meters

• Width: x meters, Length: x + 5 meters

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) = -4.9t^2 + 5. What is the factored form of this equation, and how long will it take for the ball to hit the ground?

• h(t) = -4.9(t^2 - 10), time to hit ground: ~0.71 seconds

• h(t) = -4.9(t^2 + 2.5), time to hit ground: ~1.41 seconds

• h(t) = -4.9(t^2 - 5/4.9), time to hit ground: ~1.01 seconds

• h(t) = -4.9(t^2 + 5), time to hit ground: ~2.02 seconds

The product of two consecutive integers is 72. Write a quadratic equation in factored form to represent this situation and find the integers.

• 9 and 10

• 7 and 8

• 6 and 7

• The integers are 8 and 9.

A company produces x units of a product, and the profit in dollars can be modeled by the equation P(x) = -2(x - 10)(x - 20). What is the maximum profit, and how many units should be produced to achieve it?

• Maximum profit is 30 dollars, produced at 5 units.

• Maximum profit is 60 dollars, produced at 20 units.

• Maximum profit is 40 dollars, produced at 10 units.

• Maximum profit is 50 dollars, produced at 15 units.

A rectangular pool has a length that is twice its width. If the area of the pool is 200 square meters, express the area as a quadratic equation in factored form and find the dimensions of the pool.

• Width: 15 meters, Length: 30 meters

• Width: 8 meters, Length: 16 meters

• Width: 5 meters, Length: 10 meters

• Width: 10 meters, Length: 20 meters

The height of a projectile is modeled by the equation h(t) = -16(t - 2)(t - 5). What is the maximum height reached by the projectile, and at what time does it occur?

• Maximum height: 20 feet, Time: 4 seconds

• Maximum height: 40 feet, Time: 2 seconds

• Maximum height: 32 feet, Time: 3 seconds

• Maximum height: 25 feet, Time: 5 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 60 square meters, write a quadratic equation in factored form and find the dimensions of the field.

• Width: 7 meters, Length: 11 meters

• Width: 6 meters, Length: 10 meters

• Width: 8 meters, Length: 12 meters

• Width: 5 meters, Length: 9 meters

The area of a triangular plot of land is given by the equation A = (1/2)(b)(h) where b is the base and h is the height. If the base is 3 meters longer than the height, express the area as a quadratic equation in factored form and find the dimensions of the plot.

• Height = 2 meters, Base = 5 meters

• Height = 4 meters, Base = 7 meters

• Height = 1 meter, Base = 4 meters

• Height = 3 meters, Base = 6 meters

A car's distance from a starting point can be modeled by the equation d(t) = (t - 3)(t - 7) where t is time in seconds. What is the total distance traveled by the car when it reaches its maximum distance?

• 8

• 6

• 4

• 2

A rectangular box has a volume of 120 cubic centimeters. If the length is 2 centimeters longer than the width and the height is 3 centimeters, write a quadratic equation in factored form to find the dimensions of the box.

• (x + 10)(x - 2) = 0

• (x + 8)(x - 5) = 0

• (x + 6)(x - 4) = 0

• (x + 5)(x - 3) = 0