Mathematical Reasoning and Modeling Flashcards

A town's population grows according to the model P(t) = 12,000 · (1.03)^t, where t is years. What does the value 1.03 represent?

Answer: A 3% annual growth rate

In an exponential growth model P = P₀ · b^t, the base b = 1.03 means the population increases by 3% each year.

A rectangle has a perimeter of 54 cm and a length that is twice its width. Which equation correctly models this situation?

Answer: 2w + 2(2w) = 54

Perimeter = 2(length + width) = 2(2w + w) = 6w = 54, so the equation 2w + 2(2w) = 54 correctly models the perimeter.

On a number line, which value is closest to √50?

Answer: 7

Since 7² = 49 and 8² = 64, √50 ≈ 7.07, making 7 the closest whole number.

A car rental company charges $40 per day plus $0.15 per mile. Which inequality represents renting for 3 days and staying under $200?

Answer: 120 + 0.15m ≤ 200

The cost for 3 days is 3 × $40 = $120 plus $0.15 per mile, giving 120 + 0.15m ≤ 200 for staying within budget.

A function is defined as f(x) = 3x² − 12. For which value of x does f(x) = 0?

Answer: x = ±2

Setting 3x² − 12 = 0 gives x² = 4, so x = ±2.

Two hikers start at the same point. One walks north at 4 mph and the other walks east at 3 mph. After 2 hours, how far apart are they?

Answer: 10 miles

After 2 hours, one hiker traveled 8 miles north and the other 6 miles east; by the Pythagorean theorem, distance = √(8² + 6²) = √100 = 10 miles.

A store sells notebooks for $2.50 each and pens for $1.00 each. Jake buys 4 notebooks and some pens, spending exactly $14. How many pens did he buy?

Answer: 5

4 notebooks cost $10, leaving $4 for pens; at $1.00 each, Jake bought 4 ÷ 1 = 4 pens — wait, $14 − $10 = $4, so 4 pens; correct answer is 4.