Roots and Quadratic Equations: Garden to Projectile Paths

A farmer is planting a rectangular garden. The area of the garden is represented by the equation A = x(10 - x). Create a table of values for x and A. What are the roots of the equation?

• x = 5 and x = 15

• x = -2 and x = 12

• x = 1 and x = 9

• The roots of the equation are x = 0 and x = 10.

A ball is thrown upwards, and its height in meters after t seconds is given by the equation h(t) = -5t^2 + 20t + 15. Fill in a table for t from 0 to 4 seconds. What are the roots of the equation?

• Roots: t = 1, t = 3; Table: [{"t": 0, "h(t)": 15}, {"t": 1, "h(t)": 30}, {"t": 2, "h(t)": 35}, {"t": 3, "h(t)": 30}, {"t": 4, "h(t)": 15}]

• Roots: t = 2, t = 4

• Table: [{"t": 0, "h(t)": 15}, {"t": 1, "h(t)": 20}, {"t": 2, "h(t)": 25}, {"t": 3, "h(t)": 20}, {"t": 4, "h(t)": 15}]

• Roots: t = 0, t = 4

A company's profit in thousands of dollars can be modeled by the equation P(x) = -2x^2 + 12x - 10, where x is the number of units sold. Create a table of values for x and P. Use the quadratic formula to find the break-even points.

• Break-even points are at x = 0 and x = 6.

• Break-even points are at x = 2 and x = 4.

• Break-even points are at x = -1 and x = 7.

• Break-even points are at x = 1 and x = 5.

The path of a projectile is modeled by the equation h(t) = -4.9t^2 + 20t + 5. Construct a table for t from 0 to 5 seconds. Identify the roots of the equation using the quadratic formula.

• Table: [{"t": 0, "h(t)": 10}, {"t": 1, "h(t)": 15}, {"t": 2, "h(t)": 25}, {"t": 3, "h(t)": 35}, {"t": 4, "h(t)": 45}, {"t": 5, "h(t)": 55}]

• Roots: t = 1.5 seconds and t = 3.5 seconds

• Roots: t = 0.25 seconds and t = 4.05 seconds; Table: [{"t": 0, "h(t)": 5}, {"t": 1, "h(t)": 20.1}, {"t": 2, "h(t)": 35.4}, {"t": 3, "h(t)": 50.9}, {"t": 4, "h(t)": 66.6}, {"t": 5, "h(t)": 82.5}]

• Roots: t = -1 seconds and t = 6 seconds

A rectangular swimming pool has a length that is 3 meters longer than its width. The area of the pool is 54 square meters. Set up a quadratic equation, create a table of values, and find the dimensions of the pool.

• Width: 4 meters, Length: 7 meters

• Width: 7 meters, Length: 10 meters

• Width: 6 meters, Length: 9 meters

• Width: 5 meters, Length: 8 meters

The height of a bridge is modeled by the equation h(x) = -x^2 + 6x + 8, where x is the distance from the base of the bridge. Create a table of values for x and h. What are the roots of the equation?

• The roots of the equation are 2 and 4.

• The roots of the equation are 0 and 8.

• The roots of the equation are approximately 5 and 3.

• The roots of the equation are approximately 7.123 and -1.123.

A toy rocket's height is modeled by the equation h(t) = -16t^2 + 64t + 48. Fill in a table for t from 0 to 4 seconds. Use the quadratic formula to find when the rocket hits the ground.

• The rocket hits the ground at t = 5 seconds.

• The rocket hits the ground at t = 3 seconds.

• The rocket hits the ground at t = 2.5 seconds.

• The rocket hits the ground at t = 4.5 seconds.

A rectangular garden has a perimeter of 40 meters. The length is represented as L and the width as W. If W = 10 - L, create a table of values for L and W. What are the roots of the equation for the area?

• L = 1 and L = 9

• The roots of the equation for the area are L = 0 and L = 10.

• L = 5 and L = 15

• L = 2 and L = 8

The area of a triangular plot of land is given by the equation A = 0.5 * b * h, where b is the base and h is the height. If the height is represented as h = 2b - 4, create a table of values for b and A. Identify the roots of the equation.

• Table of values: [{"b": 0, "A": 0}, {"b": 1, "A": 1}, {"b": 2, "A": 2}, {"b": 3, "A": 5}, {"b": 4, "A": 10}] Roots: b = 1, b = 3.

• Table of values: [{"b": 0, "A": 0}, {"b": 1, "A": 0.5}, {"b": 2, "A": 1}, {"b": 3, "A": 2.5}, {"b": 4, "A": 5}] Roots: b = 1, b = 3.

• Table of values: [{"b": 0, "A": 0}, {"b": 1, "A": -1}, {"b": 2, "A": 0}, {"b": 3, "A": 3}, {"b": 4, "A": 8}] Roots: b = 0, b = 2.

• Table of values: [{"b": 0, "A": 0}, {"b": 1, "A": 2}, {"b": 2, "A": 4}, {"b": 3, "A": 6}, {"b": 4, "A": 12}] Roots: b = 1, b = 4.

A car's distance from a starting point is modeled by the equation d(t) = -4.9t^2 + 20t + 10. Create a table of values for t from 0 to 5 seconds. Use the quadratic formula to find when the car returns to the starting point.

• The car never returns to the starting point.

• The car returns to the starting point at t = 6.0 seconds.

• The car returns to the starting point at t = 2.5 seconds.

• The car returns to the starting point at approximately t = 4.08 seconds.