Task Sets: Recognize constant percent rate situations - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

1.

Which situation describes exponential growth?

A.

A factory produces 500 widgets per day.

B.

A bank account earns 5% interest each year, compounded annually.

C.

A car travels 65 miles per hour.

D.

A candle loses 2 cm per hour as it burns.

2.

A quantity starts at 1,000 and grows by 6% per year. What is the value after 1 year?

3.

A function is $g(x) = 500 \cdot (1.08)^x$ . What type of function is this?

A.

Linear — the base 1.08 acts like a slope.

B.

Quadratic — the exponent makes it grow faster.

C.

Exponential — it has the form $a \cdot b^x$ with constant multiplier per period.

D.

It cannot be determined without a table.

B

Fluency Practice

1.

A town grows by 3% per year. Which function models the population $P$ after $t$ years, starting from 12,000?

A.

$P(t) = 12000 + 360t$

B.

$P(t) = 12000 \cdot (1.03)^t$

C.

$P(t) = 12000 \cdot (0.03)^t$

D.

$P(t) = 12000 + 0.03t$

2.

Which phrase signals a constant PERCENT rate situation (exponential)?

A.

A savings account earns $150 per year.

B.

A pool fills at 40 gallons per minute.

C.

A medication loses 25% of its remaining amount every hour.

D.

A store adds 50 new products each month.

3.

A car worth $18,000 depreciates 12% per year. What is the decay factor?

Enter as a decimal (e.g., 0.88).

4.

A culture of bacteria doubles every 4 hours. Starting with 500 cells, what is the growth factor per 4-hour period?

5.

An exponential function has growth factor $b = 1.07$ . What is the annual percent growth rate? Enter your answer as a whole number (e.g., 7 for 7%).

C

Mixed Practice

D

Word Problems

1.

A radioactive substance has a half-life of 6 hours. The initial amount is 400 mg.

Write the exponential function $M(t)$ for the remaining mass in mg after $t$ hours.

Enter the initial value and base as "a b" (e.g., "400 0.5" for $M(t) = 400 \cdot (0.5)^{t/6}$ ).

2.

A town had 8,500 residents in 2020 and grows at 4% per year.

1.

Write the exponential function $P(t)$ for the population $t$ years after 2020. Enter as "initial_value growth_factor" (e.g., "8500 1.04").

2.

To the nearest whole number, what is the predicted population in 2030 (when $t = 10$ )?

Use $1.04^{10} \approx 1.4802$ .

3.

Two investment accounts both start with $5,000. Account A earns $300 per year (simple interest). Account B earns 6% per year (compound interest).

In year 1, both accounts earn the same amount (since 6% of $5,000 = $300). Explain why Account B will pull ahead of Account A in future years, using the idea of constant percent rate.

E

Error Analysis

1.

A student solved the following problem:

"A population of 10,000 grows by 5% per year. Write the function."

Student work: $P(t) = 10000 + 500t$

What error did the student make?

A.

The student computed 5% of the initial value (500) and used it as a constant additive rate, treating exponential growth as linear.

B.

The student forgot to include the initial value.

C.

The student used the wrong percent — it should be 0.05, not 500.

D.

The function is correct for the first year only, but the error is in the units.

2.

A student solved the following problem:

"A quantity grows by 8% per year. Starting value is 200. Write the exponential function."

Student work: $g(t) = 200 \cdot (0.08)^t$

What error did the student make?

A.

The student forgot to include the initial value.

B.

The student used the percent rate (0.08) as the base instead of the growth factor (1.08).

C.

The student should have used a negative exponent for decay.

D.

The function is correct — 0.08 is the growth rate.

F

Challenge

1.

A substance decays at 15% per hour. A student says: "After 20 hours, the substance must be gone — 15% × 20 = 300%, which is more than 100%."

Is the student's reasoning correct? Explain why or why not, and find the actual remaining percentage after 20 hours.

2.

A quantity doubles every 3 years. If the initial value is 50, write the growth factor per YEAR (not per 3-year period).