NBT Mathematical Modelling Flashcards

A swimming pool is being filled by two pipes. Pipe A fills the pool in 6 hours and Pipe B fills it in 4 hours. How long will it take to fill the pool if both pipes are open?

Answer: 2.4 h

Combined rate = 1/6 + 1/4 = 5/12 pools per hour; time = 12/5 = 2.4 hours.

A scientist models the cooling of a substance with T(t) = 20 + 80e^(−0.5t), where T is temperature (°C) and t is time (hours). What is the temperature at t = 0?

Answer: 100°C

At t = 0, e^0 = 1, so T(0) = 20 + 80(1) = 100°C.

A farmer encloses a rectangular plot using 120 feet of fencing, with one side against a barn (no fence needed). What width maximizes the enclosed area?

Answer: 30 ft

With one side free, 2w + l = 120; Area = w(120 − 2w) is maximized at w = 30 ft.

The value of a car depreciates by 15% each year. If the car initially costs $24,000, which expression gives its value after n years?

Answer: 24000(0.85)ⁿ

Each year the car retains 85% of its value, so value = 24000(0.85)ⁿ.

A company's profit P (in thousands) is modeled by P = −2x² + 20x − 30, where x is units sold (hundreds). How many units (in hundreds) maximizes profit?

Answer: 5

Vertex at x = −20/(2×−2) = 5; so 500 units maximizes profit.

A map uses a scale of 1 inch : 50 miles. Two cities are 3.6 inches apart on the map. A driver travels at 60 mph. How long will the trip take?

Answer: 3 h

Actual distance = 3.6 × 50 = 180 miles; time = 180 ÷ 60 = 3 hours.

A function f(x) = 3x − 7 models a manufacturing output. If f(x) = 20, what is the input value x?

Answer: 9

Solving 3x − 7 = 20 gives 3x = 27, so x = 9.