Master Quadratic Functions: Applications in Real-World Problems

Question 1

A spherical balloon is being inflated with air at a constant rate of 10 cubic centimeters per second. After 10 seconds, what is the radius of the balloon? (Volume of a sphere: V = (4/3)πr³)

Accepted Answer

(100π/3)^(1/3) cm

Answer

10 cm

Answer

10π cm

Answer

100π cm

Question 2

A ball is launched vertically upwards with an initial velocity of 15 m/s. Calculate the maximum height that the ball will attain.

Accepted Answer

22.5 meters

Answer

11.25 meters

Answer

33.75 meters

Answer

45 meters

Question 3

The vertex of a quadratic function is located at (-3, 2). Determine the axis of symmetry of the function.

Accepted Answer

x = -3

Answer

x = -2

Answer

x = 2

Answer

x = 3

Question 4

Which of the following is the equation of a quadratic function with a maximum value of 4 at x = 2?

Accepted Answer

y = -2(x - 2)^2 + 4

Answer

y = -2(x + 2)^2 + 4

Answer

y = 2(x - 2)^2 + 4

Answer

y = 2(x + 2)^2 + 4

Question 5

Calculate the discriminant of the quadratic function f(x) = 2x^2 - 5x + 3.

Accepted Answer

9

Answer

1

Answer

17

Answer

25

Question 6

The graph of a quadratic function passes through the points (1, -3), (2, 0), and (3, 3). Determine the equation of the function.

Accepted Answer

y = x^2 - 2x - 3

Answer

y = x^2 + 2x - 3

Answer

y = -x^2 + 2x - 3

Answer

y = -x^2 - 2x - 3

Question 7

A company estimates that its profit (in pounds) from selling x units of a product is given by the function P(x) = -0.01x^2 + 10x - 50. How many items must be sold to maximize profit?

Accepted Answer

500 items

Answer

1000 items

Answer

750 items

Answer

250 items

Question 8

What is the discriminant of a quadratic equation in the form of ax^2 + bx + c = 0?

Accepted Answer

b^2 - 4ac

Answer

4a^2 - b^2

Answer

a^2 - 4bc

Answer

b^2 + 4ac

Question 9

Which of the following is the equation of the axis of symmetry of the parabola y = x^2 - 4x + 3?

Accepted Answer

x = 2

Answer

x = 3

Answer

x = -2

Answer

x = 1

Question 10

Which of the following methods can be used to solve a quadratic equation?

Accepted Answer

Factoring, completing the square, quadratic formula

Answer

Integration, differentiation, logarithms

Answer

Substitution, elimination, graphing

Question 11

What is the range of the function y = (x - 3)^2 - 2?

Accepted Answer

[-2, ∞)

Answer

(-∞, ∞)

Answer

[-3, ∞)

Answer

[-2, 3]

Question 12

Which of the following graphs represents a quadratic function that has a positive leading coefficient?

Accepted Answer

A parabola opening upwards

Answer

A straight line with a negative slope

Answer

A parabola opening downwards

Answer

A straight line with a positive slope

Question 13

A ball is thrown vertically upwards with an initial velocity of 15 m/s. Its height, h, after t seconds is given by the equation h = 15t - 5t². What is the maximum height reached by the ball?

Accepted Answer

11.25 m

Answer

7.5 m

Answer

30 m

Answer

15 m

Question 14

The path of a projectile is modeled by the equation y = -x² + 4x, where y is the height in meters and x is the horizontal distance in meters. What is the maximum height reached by the projectile?

Accepted Answer

4 m

Answer

2 m

Answer

8 m

Answer

16 m

Question 15

A rectangular garden is to be enclosed by 20 metres of fencing. What is the maximum area the garden can have?

Accepted Answer

25 square metres

Answer

100 square metres

Answer

50 square metres

Answer

10 square metres

Question 16

A company produces x units of a product. The cost of production is given by the equation C = 2x² - 10x + 50. At what production level is the cost minimized?

Accepted Answer

x = 2.5 units

Answer

x = 1 unit

Answer

x = 5 units

Question 17

A projectile is launched from ground level with an initial velocity of 30 m/s at an angle of 45 degrees to the horizontal. The horizontal distance travelled by the projectile is given by the equation d = (v² sin 2θ)/g, where v is the initial velocity, θ is the launch angle, and g is the acceleration due to gravity (9.8 m/s²). What is the maximum horizontal distance the projectile can travel?

Accepted Answer

91.8 metres

Answer

45.9 metres

Answer

183.6 metres

Answer

30 metres

Question 18

The height, h, of a ball thrown vertically upwards is given by the equation h = -4.9t² + 20t + 1, where t is the time in seconds. After how many seconds does the ball reach its maximum height?

Accepted Answer

t = 2.04 seconds (approximately)

Answer

t = 4 seconds

Answer

t = 1 second

Question 19

The number of bacteria in a culture is given by the equation N = 1000t² - 500t + 100, where t is the time in hours. When does the bacterial population reach its minimum value?

Accepted Answer

t = 0.25 hours

Answer

t = 1 hour

Answer

t = 2 hours

Answer

t = 5 hours

Question 20

A rectangular garden is to be enclosed by 50 metres of fencing. One side of the garden is to be twice the length of the other. What are the dimensions of the garden that maximize its area?

Accepted Answer

Length = 16.67 metres, Width = 8.33 metres (approximately)

Answer

Length = 10 metres, Width = 5 metres

Answer

Length = 25 metres, Width = 12.5 metres

Question 21

A company produces and sells widgets. The cost function for producing x widgets is given by C(x) = 0.5x² + 10x + 500, and the revenue function is given by R(x) = 20x. At what production level does the company break even?

Accepted Answer

x = 20

Answer

x = 10

Answer

x = 30

Answer

x = 40

Question 22

The height of a projectile launched from a cliff is given by the equation h(t) = -5t² + 20t + 100, where t is time in seconds. What is the initial height of the projectile?

Accepted Answer

100 meters

Answer

5 meters

Answer

0 meters

Answer

20 meters

Question 23

A manufacturer finds that the cost, C (in pounds), of producing x items is given by the function C(x) = 0.01x² + 5x + 100. What is the minimum cost of production?

Accepted Answer

£225

Answer

£100

Answer

£500

Answer

£125