Mastering Graphs & Systems of Nonlinear Equations

A farmer has a rectangular field and a circular pond. The area of the field is represented by the equation A = x^2 + 4x, and the area of the pond is given by A = πr^2. If the total area of the field and pond is 1000 square meters, find the dimensions of the field and the radius of the pond. Graph the equations to visualize the solution.

• Field dimensions: 15m x 30m; Pond radius: ~10m

• Field dimensions: 20m x 24m; Pond radius: ~13.82m

• Field dimensions: 30m x 10m; Pond radius: ~8m

• Field dimensions: 25m x 20m; Pond radius: ~5m

A company produces two types of products, A and B. The profit from product A is modeled by the equation P_A = -2x^2 + 12x, and from product B by P_B = -x^2 + 10x. If the total profit from both products is $50, determine the number of each product that should be produced to maximize profit. Solve the system algebraically and graph the equations.

• Produce 6 units of product A and 4 units of product B.

• Produce 7 units of product A and 3 units of product B.

• Produce 4 units of product A and 6 units of product B.

• Produce 5 units of product A and 5 units of product B.

A car rental company charges a flat fee of $30 plus $0.20 per mile driven. Another company charges a flat fee of $20 plus $0.25 per mile. Determine the number of miles at which both companies charge the same amount. Graph the equations to find the intersection point.

• 200 miles

• 100 miles

• 250 miles

• 150 miles

A ball is thrown from the top of a 50-meter building. The height of the ball can be modeled by the equation h(t) = -4.9t^2 + 50, while the height of a second ball thrown from the ground is modeled by h(t) = 3t + 5. Find the time at which both balls are at the same height and graph the equations to illustrate the solution.

• t ≈ 4.5 seconds

• t ≈ 2.0 seconds

• t ≈ 6.3 seconds

• t ≈ 3.1 seconds

A rectangular garden is to be designed with a fixed perimeter of 60 meters. The area of the garden can be represented by the equation A = x(30 - x), where x is the length of one side. If a circular fountain with a radius of r meters is to be placed in the garden, find the dimensions of the garden and the radius of the fountain such that the total area is maximized. Solve the system algebraically and graph the equations.

• Dimensions of the garden: 15m x 15m; Radius of the fountain: up to 7.5m

• Dimensions of the garden: 20m x 10m; Radius of the fountain: up to 8m

• Dimensions of the garden: 10m x 20m; Radius of the fountain: up to 5m

• Dimensions of the garden: 12m x 18m; Radius of the fountain: up to 6m

A local theater sells tickets for two different shows. The revenue from show A is modeled by R_A = 50x - 2x^2, and from show B by R_B = 40y - y^2. If the total revenue from both shows is $5000, determine the number of tickets sold for each show. Graph the equations to find the solution.

• x = 30, y = 60

• x = 25, y = 75

• x = 20, y = 80

• x = 40, y = 50

A cyclist travels 10 km at a speed of x km/h and returns at a speed of y km/h. The total time for the trip is 1 hour. If the relationship between the speeds is given by y = 2x - 5, find the speeds of the cyclist for both trips. Solve the system algebraically and graph the equations to visualize the solution.

• x = 5 km/h, y = 10 km/h

• x = 8 km/h, y = 12 km/h

• x = 10 km/h, y = 15 km/h

• x = 12 km/h, y = 20 km/h

A rectangular swimming pool has a length that is 3 meters longer than its width. The area of the pool is 150 square meters. If a circular hot tub with a radius of r meters is to be placed next to the pool, find the dimensions of the pool and the radius of the hot tub such that the total area is maximized. Solve the system algebraically and graph the equations.

• Width: 8 meters, Length: 11 meters, Radius of hot tub: 2 meters

• Width: 12 meters, Length: 15 meters, Radius of hot tub: 3 meters

• Width: 9 meters, Length: 12 meters, Radius of hot tub: 1 meter

• Width: 10.45 meters, Length: 13.45 meters, Radius of hot tub: r (to be determined based on additional constraints)

A store sells two types of fruit baskets. The cost of basket A is modeled by C_A = 3x + 2y, and the cost of basket B by C_B = 4x + y. If the total cost for both baskets is $100, determine the number of each type of basket that can be purchased. Graph the equations to find the solution.

• (11, -1)

• (0, 11)

• Possible solutions include (10, 0), (9, 1), (8, 2), (7, 3), (6, 4), (5, 5), (4, 6), (3, 7), (2, 8), (1, 9), (0, 10).

• (5, 10)

A drone can fly at a height modeled by the equation h(t) = -5t^2 + 20t + 10, while a balloon rises at a constant rate modeled by h(t) = 15 + 2t. Find the time at which both the drone and the balloon are at the same height. Solve the system algebraically and graph the equations to illustrate the solution.

• t = 5 seconds and t = 6 seconds

• t = 1 seconds and t = 2 seconds

• t = 1.5 seconds and t = 4 seconds

• t = 0 seconds and t = 3 seconds