Quadratics in Housing: Finding Dimensions & Identifying Vertices

A rectangular garden is designed with a length that is 3 meters longer than its width. If the area of the garden is 54 square meters, what are the dimensions of the garden? Identify the vertex of the quadratic equation formed.

• Width: 5 meters, Length: 8 meters

• Width: 7 meters, Length: 10 meters

• Width: 4 meters, Length: 7 meters

• Width: 6 meters, Length: 9 meters

A house's value can be modeled by the equation V(t) = -2t^2 + 12t + 50, where V is the value in thousands of dollars and t is the number of years since it was built. What is the maximum value of the house, and when does it occur?

• The maximum value is 60 thousand dollars, occurring at 2 years.

• The maximum value is 72 thousand dollars, occurring at 4 years.

• The maximum value is 50 thousand dollars, occurring at 0 years.

• The maximum value is 68 thousand dollars, occurring at 3 years.

A homeowner wants to build a fence around a rectangular yard. The length of the yard is 4 meters more than twice the width. If the area of the yard is 96 square meters, find the dimensions of the yard and identify the axis of symmetry of the quadratic equation.

• Width: 10 meters, Length: 24 meters, Axis of symmetry: -2

• Width: 6 meters, Length: 16 meters, Axis of symmetry: -1

• Width: 8 meters, Length: 20 meters, Axis of symmetry: 0

• Width: 4 meters, Length: 12 meters, Axis of symmetry: 2

The height of a projectile launched from a house can be modeled by the equation h(t) = -5t^2 + 20t + 15, where h is the height in meters and t is the time in seconds. What is the maximum height reached by the projectile?

• 25 meters

• 35 meters

• 30 meters

• 40 meters

A triangular plot of land has a base that is 2 meters longer than its height. If the area of the plot is 30 square meters, find the height and base of the triangle. What quadratic equation do you form, and what is its vertex?

• Height = 5 meters, Base = 7 meters

• Height = 3 meters, Base = 5 meters

• Height = 6 meters, Base = 8 meters

• Height = 4 meters, Base = 6 meters

A swimming pool is being designed in a backyard. The length is 2 meters longer than the width, and the area is 48 square meters. Write the quadratic equation for the area and find the dimensions of the pool. What is the axis of symmetry?

• Width: 6 meters, Length: 8 meters, Axis of symmetry: -1

• Width: 10 meters, Length: 12 meters, Axis of symmetry: -2

• Width: 4 meters, Length: 6 meters, Axis of symmetry: 2

• Width: 5 meters, Length: 7 meters, Axis of symmetry: 0

A local builder finds that the cost C (in thousands of dollars) to build a house can be modeled by the equation C(x) = 3x^2 - 12x + 15, where x is the number of houses built. What is the minimum cost, and how many houses should be built to achieve this cost?

• Minimum cost is 5 thousand dollars when 3 houses are built.

• Minimum cost is 4 thousand dollars when 4 houses are built.

• Minimum cost is 2 thousand dollars when 1 house is built.

• Minimum cost is 3 thousand dollars when 2 houses are built.

A rectangular lot has a length that is 5 meters more than its width. If the area of the lot is 100 square meters, find the dimensions of the lot and identify the vertex of the quadratic equation formed.

• Width: 12 meters, Length: 17 meters, Vertex: (-3, -150)

• Width: 5 meters, Length: 10 meters, Vertex: (-1, -50)

• Width: 10 meters, Length: 15 meters, Vertex: (-2.5, -112.5)

• Width: 8 meters, Length: 13 meters, Vertex: (-2, -80)

A gardener is planting a flower bed in the shape of a rectangle. The length is 3 meters more than twice the width. If the area of the flower bed is 45 square meters, find the dimensions and the axis of symmetry of the quadratic equation.

• Width: 6 meters, Length: 15 meters, Axis of symmetry: -3/4

• Width: 7 meters, Length: 18 meters, Axis of symmetry: -2

• Width: 5 meters, Length: 12 meters, Axis of symmetry: -1

• Width: 4 meters, Length: 10 meters, Axis of symmetry: -5/4

A family is planning to build a new house. The area of the house can be represented by the equation A(x) = x^2 + 6x + 8, where A is the area in square meters and x is the length of one side. What is the maximum area, and what are the dimensions of the house?

• Maximum area: 0 square meters; Dimensions: 0 meters

• Maximum area: 100 square meters; Dimensions: 10 meters

• Maximum area: 50 square meters; Dimensions: 5 meters

• Maximum area: 20 square meters; Dimensions: 4 meters