NBT NBT Mathematical Modelling 5 — Questions and Answers
Question 1: 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?
- 2 h
- 2.4 h (Correct answer)
- 3 h
- 5 h
Correct answer: 2.4 h
Combined rate = 1/6 + 1/4 = 5/12 pools per hour; time = 12/5 = 2.4 hours.
Question 2: 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?
- 20°C
- 80°C
- 100°C (Correct answer)
- 160°C
Correct answer: 100°C
At t = 0, e^0 = 1, so T(0) = 20 + 80(1) = 100°C.
Question 3: 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?
- 20 ft
- 25 ft
- 30 ft (Correct answer)
- 40 ft
Correct answer: 30 ft
With one side free, 2w + l = 120; Area = w(120 − 2w) is maximized at w = 30 ft.
Question 4: 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?
- 24000(0.15)ⁿ
- 24000(1.15)ⁿ
- 24000(0.85)ⁿ (Correct answer)
- 24000 − 15n
Correct answer: 24000(0.85)ⁿ
Each year the car retains 85% of its value, so value = 24000(0.85)ⁿ.
Question 5: 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?
- 4
- 5 (Correct answer)
- 6
- 10
Correct answer: 5
Vertex at x = −20/(2×−2) = 5; so 500 units maximizes profit.
Question 6: 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?
- 2 h
- 2.5 h
- 3 h (Correct answer)
- 4 h
Correct answer: 3 h
Actual distance = 3.6 × 50 = 180 miles; time = 180 ÷ 60 = 3 hours.
Question 7: A function f(x) = 3x − 7 models a manufacturing output. If f(x) = 20, what is the input value x?
- 7
- 8
- 9 (Correct answer)
- 10
Correct answer: 9
Solving 3x − 7 = 20 gives 3x = 27, so x = 9.
NBT NBT Mathematical Modelling 5
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?