Grade 8 Polynomial Problems: Solving & Graphing Skills

A rectangular garden has a length that is 3 meters longer than its width. If the area of the garden is represented by the polynomial equation A = x(x + 3), where x is the width, find the dimensions of the garden.

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

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

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

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

The height of a ball thrown into the air can be modeled by the polynomial function h(t) = -4.9t^2 + 20t + 1, where h is the height in meters and t is the time in seconds. How long will it take for the ball to hit the ground?

• 4.1 seconds

• 2.8 seconds

• 5.0 seconds

• 3.5 seconds

A company produces x units of a product, and the profit in dollars can be modeled by the polynomial P(x) = -2x^2 + 40x - 100. Determine the number of units that must be sold to maximize profit.

• 10

• 5

• 20

• 15

The volume of a rectangular box is given by the polynomial V = x^3 + 2x^2 - 5x, where x is the length of one side. Find the value of x that maximizes the volume of the box.

• 4.0

• 2.5

• 3.0

• 1.5

A car's distance from a starting point can be modeled by the polynomial function d(t) = 3t^3 - 12t^2 + 9t, where d is the distance in meters and t is the time in seconds. Graph this polynomial function and identify the time when the car is at rest.

• t = 2 seconds

• t = 0 seconds

• t = 5 seconds

• The car is at rest at t = (4 + √7)/3 seconds and t = (4 - √7)/3 seconds.

The cost of producing x items is given by the polynomial C(x) = 5x^2 + 20x + 100. If the company wants to minimize costs, what is the optimal number of items to produce?

• 0

• -1

• 5

• 10

A farmer has a field in the shape of a triangle with a base represented by the polynomial b(x) = 2x + 4 and a height represented by h(x) = x - 1. Write a polynomial expression for the area of the field and find the maximum area.

• 3.5

• 0.5

• -2.25

• -1.75

The profit function for a small business is given by P(x) = -x^2 + 10x + 24. Determine the x-value that maximizes the profit and explain its significance in a real-world context.

• 5

• 10

• 7

• 3

A swimming pool is being filled with water, and the volume of water in the pool can be modeled by the polynomial V(t) = 2t^3 - 15t^2 + 36t, where V is the volume in liters and t is the time in hours. Find the time when the pool is full.

• The pool can hold 1000 liters.

• The pool is full at 10 hours.

• The pool is never full.

• The pool fills completely in 5 hours.

A toy company finds that the demand for their toy can be modeled by the polynomial D(x) = -x^2 + 30x - 50, where D is the demand and x is the price. Determine the price that maximizes demand and explain how this can help the company.

• 20

• 25

• 15

• 10