11th Grade Polynomial Roots & Graphing Challenge

A farmer has a rectangular field with a length that is 3 meters longer than twice its width. If the area of the field is 150 square meters, find the dimensions of the field by solving the polynomial equation derived from the area formula.

• Width: 5 meters, Length: 15 meters

• Width: 10 meters, Length: 23 meters

• Width: 8 meters, Length: 19 meters

• Width: 7.5 meters, Length: 18 meters

A toy company produces a new model of a toy car. The profit, P, in dollars, from selling x units of the toy car is given by the polynomial P(x) = -2x^2 + 40x - 150. Determine the number of units that must be sold to maximize profit and identify the maximum profit.

• 5 units, Maximum profit is $30.

• 15 units, Maximum profit is $70.

• 20 units, Maximum profit is $100.

• 10 units, Maximum profit is $50.

A rectangular garden has a length that is 4 meters longer than its width. If the area of the garden is 96 square meters, write a polynomial equation to find the dimensions of the garden and identify the roots of the equation.

• Width: 5 meters, Length: 9 meters

• Width: 8 meters, Length: 12 meters

• Width: 6 meters, Length: 10 meters

• Width: 10 meters, Length: 14 meters

A local theater sells tickets for a play. The revenue, R, in dollars, from selling x tickets is modeled by the polynomial R(x) = -5x^2 + 200x. How many tickets need to be sold to achieve maximum revenue, and what is that maximum revenue?

• 20 tickets, $2000

• 15 tickets, $1500

• 25 tickets, $2500

• 30 tickets, $3000

A swimming pool is being filled with water. The volume of water in the pool, V, in cubic meters, is modeled by the polynomial V(t) = 2t^3 - 12t^2 + 18t, where t is the time in hours. Determine the time when the pool will be full by finding the roots of the polynomial.

• 4 hours

• 5 hours

• 2 hours

• 3 hours

A company’s production cost, C, in dollars, for producing x items is given by the polynomial C(x) = 3x^3 - 15x^2 + 24x + 10. Find the number of items produced that results in zero profit by identifying the roots of the cost equation.

• 2, 6

• 1.5, 4.5

• 3, 7

• 0.56, 5.44

A rectangular box has a volume of 120 cubic centimeters. The length is twice the width, and the height is 3 centimeters. Write a polynomial equation to find the width of the box and solve for its dimensions.

• Width: 1 cm, Length: 2 cm, Height: 3 cm

• Width: 3 cm, Length: 6 cm, Height: 3 cm

• Width: 2√5 cm, Length: 4√5 cm, Height: 3 cm

• Width: 5 cm, Length: 10 cm, Height: 3 cm

A car rental company charges a flat fee of $50 plus $20 for each day the car is rented. Write a polynomial function to represent the total cost, C, for renting a car for d days. Determine the cost for renting the car for 5 days and identify the polynomial roots if the cost is set to $150.

• The cost for renting the car for 5 days is $150, and the polynomial root is d = 10.

• The cost for renting the car for 5 days is $150, and the polynomial root is d = 5.

• The cost for renting the car for 5 days is $100, and the polynomial root is d = 3.

• The cost for renting the car for 5 days is $200, and the polynomial root is d = 7.

A school is planning a fundraiser and sells boxes of cookies. The profit, P, in dollars, from selling x boxes is modeled by the polynomial P(x) = -x^2 + 30x - 200. Find the number of boxes that need to be sold to break even by identifying the roots of the profit equation.

• 15

• 20

• 30

• 25

A rectangular plot of land has a perimeter of 100 meters. The length is 10 meters more than the width. Write a polynomial equation to find the dimensions of the plot and identify the roots of the equation to determine the width.

• 20 meters

• 25 meters

• 15 meters

• 30 meters