Solving Real-Life Problems with Substitution Methods

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 using substitution. Interpret the solutions in the context of the problem.

• Field dimensions: 25m x 20m; Pond radius: sqrt(300/π) meters.

• Field dimensions: 20m x 24m; Pond radius: sqrt(760/π) meters.

• Field dimensions: 10m x 40m; Pond radius: sqrt(900/π) meters.

• Field dimensions: 15m x 30m; Pond radius: sqrt(500/π) meters.

A car rental company charges a flat fee of $50 plus $0.20 per mile driven. Another company charges a flat fee of $30 plus $0.25 per mile. Set up a system of equations to find out how many miles you need to drive for both companies to cost the same. Use substitution to solve and interpret the result.

• 300 miles

• 200 miles

• 500 miles

• 400 miles

A rectangular garden has a length that is 3 meters longer than its width. The area of the garden is 54 square meters. Write a system of equations to represent this situation, solve using substitution, and interpret the dimensions of the garden in context.

• Width: 5 meters, Length: 8 meters

• Width: 7 meters, Length: 10 meters

• Width: 6 meters, Length: 9 meters

• Width: 4 meters, Length: 7 meters

Two friends are running a race. The first friend runs at a speed of 5 meters per second, while the second friend runs at a speed of 3 meters per second. If the first friend has a 10-meter head start, how long will it take for the second friend to catch up? Set up a system of equations and solve using substitution, then interpret the time in the context of the race.

• 5 seconds

• 3 seconds

• 10 seconds

• 7 seconds

A company produces two types of gadgets. The profit from the first type is represented by the equation P1 = 2x^2 + 3x, and the profit from the second type is P2 = 4y^2 + 5y. If the total profit is $1000, find the number of each type of gadget produced using substitution and interpret the results in terms of production.

• x = 5, y = 10

• x = 10, y = 5

• x = 15, y = 2

• x = 0, y = 20

A swimming pool is being filled with water. The rate of water flow into the pool is represented by the equation V = 2t^2 + 3t, while the rate of water evaporation is given by V = t^2 + 5t. If the pool starts with 1000 liters of water, find the time when the pool will be full using substitution and interpret the solution in the context of the pool's operation.

• The pool will be full at 2000 liters.

• The pool will never be full.

• The pool will be full in 5 hours.

• The pool will overflow after 10 hours.

A rectangular piece of land has a length that is twice its width. If the area of the land is 200 square meters, set up a system of equations to find the dimensions of the land. Use substitution to solve and interpret the dimensions in the context of land use.

• Width: 8 meters, Length: 16 meters

• Width: 15 meters, Length: 30 meters

• Width: 10 meters, Length: 20 meters

• Width: 5 meters, Length: 15 meters

A bicycle and a car travel the same distance. The bicycle travels at a speed of 15 km/h, while the car travels at a speed of 60 km/h. If the car leaves 30 minutes after the bicycle, set up a system of equations to find the distance traveled when they meet. Use substitution to solve and interpret the results in the context of their travel.

• 5 km

• 15 km

• 10 km

• 20 km

A rectangular box has a volume of 120 cubic centimeters. The length is twice the width, and the height is 3 centimeters less than the width. Set up a system of equations to find the dimensions of the box using substitution and interpret the dimensions in the context of packaging.

• Width: 4 cm, Length: 8 cm, Height: 1 cm

• Width: 7 cm, Length: 14 cm, Height: 4 cm

• Width: 5 cm, Length: 10 cm, Height: 2 cm

• Width: 6 cm, Length: 12 cm, Height: 3 cm

A local bakery sells two types of cakes. The profit from selling chocolate cakes is represented by the equation P1 = 3x^2 + 2x, and the profit from vanilla cakes is P2 = 4y^2 + 3y. If the total profit is $500, find the number of each type of cake sold using substitution and interpret the results in terms of sales.

• Chocolate cakes sold: 8, Vanilla cakes sold: 7

• Chocolate cakes sold: 12, Vanilla cakes sold: 3

• Chocolate cakes sold: 5, Vanilla cakes sold: 10

• Chocolate cakes sold: 10, Vanilla cakes sold: 5