Mathematical Reasoning and Modeling Flashcards

Which situation can best be modeled by a linear function?

Answer: The distance traveled by a car moving at a constant speed

Constant speed means equal distance is covered each hour, producing a linear relationship between time and distance.

The table shows: x = 1,2,3,4 and y = 5,8,11,14. What is the equation of the linear function?

Answer: y = 3x + 2

The rate of change is (8−5)/(2−1) = 3, and using the point (1,5): 5 = 3(1) + b gives b = 2, so y = 3x + 2.

A scale model of a building uses a ratio of 1:200. If the model is 15 cm tall, how tall is the actual building in meters?

Answer: 30 m

15 cm × 200 = 3,000 cm = 30 meters.

A quadratic equation has roots at x = −3 and x = 5. Which equation represents this?

Answer: (x + 3)(x − 5) = 0

Roots at x = −3 and x = 5 come from factors (x + 3) = 0 and (x − 5) = 0.

Maria earns $15 per hour and works h hours. Her rent is $900 per month. Which inequality models how many hours she must work to afford rent?

Answer: 15h ≥ 900

She needs her earnings (15h) to be at least $900, so 15h ≥ 900.

A graph of y = f(x) passes through (0, 3) and has a slope of −2. What is f(5)?

Answer: −7

Using y = −2x + 3: f(5) = −2(5) + 3 = −10 + 3 = −7.

The volume of a sphere is given by V = (4/3)πr³. If the radius is doubled, by what factor does the volume increase?

Answer: 8

Replacing r with 2r gives V = (4/3)π(2r)³ = (4/3)π·8r³, which is 8 times the original volume.