Exponential growth and decay (dy/dt = ky) worksheet

A quantity that changes at a rate proportional to its size satisfies dy/dt = ky, and the solution is always y = y(0)e^(kt). The work is finding k from what you are told, then reading the model forwards (how much later) or backwards (how long until).

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Sigma Prep · sigmaprep.io/worksheets Name Growth and Decay: dy/dt = ky Date Period Answer each question. Give exact answers unless the question says to round. A population doubles every $2$ years and $P (0) = 400$ . It changes at a rate proportional to its size. Write $P (t)$ . $\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (10) = 150$ . Find $k$ exactly. A quantity follows $\frac{d y}{d t} = - 0.06 y$ . Find its half-life, to the nearest tenth. Solve $\frac{d y}{d t} = - 0.08 y$ , $y (1) = 250$ . $\frac{d y}{d t} = k y$ , with $y (0) = 100$ and $y (3) = 125$ . Find $k$ exactly. A substance has a half-life of $3$ years. Starting from $1000$ g, how much is left after $10$ years? Round to the nearest tenth. A substance has a half-life of $6$ years and $N (0) = 1200$ . It changes at a rate proportional to its size. Write $N (t)$ . $\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (5) = 450$ . Find $k$ exactly. A quantity follows $\frac{d y}{d t} = 0.1 y$ . Find its doubling time, to the nearest tenth. Solve $\frac{d y}{d t} = 0.04 y$ , $y (3) = 50$ . $\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (2) = 50$ . Find $k$ exactly. A substance has a half-life of $3$ years. Starting from $1200$ g, how much is left after $32$ years? Round to the nearest tenth. A population doubles every $3$ years and $P (0) = 150$ . It changes at a rate proportional to its size. Write $P (t)$ . $\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (7) = 750$ . Find $k$ exactly. A substance has a half-life of $10$ years. Starting from $1500$ g, how much is left after $37$ years? Round to the nearest tenth. A substance has a half-life of $10$ years and $N (0) = 600$ . It changes at a rate proportional to its size. Write $N (t)$ . $\frac{d y}{d t} = k y$ , with $y (0) = 50$ and $y (5) = 40$ . Find $k$ exactly. A substance has a half-life of $3$ years. Starting from $50$ g, how much is left after $35$ years? Round to the nearest tenth. Solve $\frac{d y}{d t} = 0.02 y$ , $y (4) = 1200$ . $\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (2) = 1500$ . Find $k$ exactly. Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Growth and Decay: dy/dt = ky

Date Period

Answer each question. Give exact answers unless the question says to round.

A population doubles every $2$ years and $P (0) = 400$ . It changes at a rate proportional to its size. Write $P (t)$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (10) = 150$ . Find $k$ exactly.
A quantity follows $\frac{d y}{d t} = - 0.06 y$ . Find its half-life, to the nearest tenth.
Solve $\frac{d y}{d t} = - 0.08 y$ , $y (1) = 250$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 100$ and $y (3) = 125$ . Find $k$ exactly.
A substance has a half-life of $3$ years. Starting from $1000$ g, how much is left after $10$ years? Round to the nearest tenth.
A substance has a half-life of $6$ years and $N (0) = 1200$ . It changes at a rate proportional to its size. Write $N (t)$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (5) = 450$ . Find $k$ exactly.
A quantity follows $\frac{d y}{d t} = 0.1 y$ . Find its doubling time, to the nearest tenth.
Solve $\frac{d y}{d t} = 0.04 y$ , $y (3) = 50$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (2) = 50$ . Find $k$ exactly.
A substance has a half-life of $3$ years. Starting from $1200$ g, how much is left after $32$ years? Round to the nearest tenth.
A population doubles every $3$ years and $P (0) = 150$ . It changes at a rate proportional to its size. Write $P (t)$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (7) = 750$ . Find $k$ exactly.
A substance has a half-life of $10$ years. Starting from $1500$ g, how much is left after $37$ years? Round to the nearest tenth.
A substance has a half-life of $10$ years and $N (0) = 600$ . It changes at a rate proportional to its size. Write $N (t)$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 50$ and $y (5) = 40$ . Find $k$ exactly.
A substance has a half-life of $3$ years. Starting from $50$ g, how much is left after $35$ years? Round to the nearest tenth.
Solve $\frac{d y}{d t} = 0.02 y$ , $y (4) = 1200$ .
$\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (2) = 1500$ . Find $k$ exactly.



Sigma Prep · sigmaprep.io/worksheets Name Growth and Decay: dy/dt = kyAnswer keyVersion 1 Date Period Answer each question. Give exact answers unless the question says to round. A population doubles every $2$ years and $P (0) = 400$ . It changes at a rate proportional to its size. Write $P (t)$ . $P (t) = 400 \cdot 2^{t / 2}$ $\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (10) = 150$ . Find $k$ exactly. $k = - \frac{\ln 4}{10}$ A quantity follows $\frac{d y}{d t} = - 0.06 y$ . Find its half-life, to the nearest tenth. $11.6$ Solve $\frac{d y}{d t} = - 0.08 y$ , $y (1) = 250$ . $y (t) = 250 e^{- 0.08 (t - 1)}$ $\frac{d y}{d t} = k y$ , with $y (0) = 100$ and $y (3) = 125$ . Find $k$ exactly. $k = \frac{1}{3} \ln ⁣ (\frac{5}{4})$ A substance has a half-life of $3$ years. Starting from $1000$ g, how much is left after $10$ years? Round to the nearest tenth. $99.2$ g A substance has a half-life of $6$ years and $N (0) = 1200$ . It changes at a rate proportional to its size. Write $N (t)$ . $N (t) = 1200 \cdot {(\frac{1}{2})}^{t / 6}$ $\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (5) = 450$ . Find $k$ exactly. $k = \frac{1}{5} \ln ⁣ (\frac{3}{2})$ A quantity follows $\frac{d y}{d t} = 0.1 y$ . Find its doubling time, to the nearest tenth. $6.9$ Solve $\frac{d y}{d t} = 0.04 y$ , $y (3) = 50$ . $y (t) = 50 e^{0.04 (t - 3)}$ $\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (2) = 50$ . Find $k$ exactly. $k = - \frac{\ln 4}{2}$ A substance has a half-life of $3$ years. Starting from $1200$ g, how much is left after $32$ years? Round to the nearest tenth. $0.7$ g A population doubles every $3$ years and $P (0) = 150$ . It changes at a rate proportional to its size. Write $P (t)$ . $P (t) = 150 \cdot 2^{t / 3}$ $\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (7) = 750$ . Find $k$ exactly. $k = \frac{1}{7} \ln ⁣ (\frac{5}{2})$ A substance has a half-life of $10$ years. Starting from $1500$ g, how much is left after $37$ years? Round to the nearest tenth. $115.4$ g A substance has a half-life of $10$ years and $N (0) = 600$ . It changes at a rate proportional to its size. Write $N (t)$ . $N (t) = 600 \cdot {(\frac{1}{2})}^{t / 10}$ $\frac{d y}{d t} = k y$ , with $y (0) = 50$ and $y (5) = 40$ . Find $k$ exactly. $k = - \frac{1}{5} \ln ⁣ (\frac{5}{4})$ A substance has a half-life of $3$ years. Starting from $50$ g, how much is left after $35$ years? Round to the nearest tenth. $0.0$ g Solve $\frac{d y}{d t} = 0.02 y$ , $y (4) = 1200$ . $y (t) = 1200 e^{0.02 (t - 4)}$ $\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (2) = 1500$ . Find $k$ exactly. $k = \frac{1}{2} \ln ⁣ (\frac{5}{2})$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Growth and Decay: dy/dt = kyAnswer keyVersion 1

Date Period

Answer each question. Give exact answers unless the question says to round.

A population doubles every $2$ years and $P (0) = 400$ . It changes at a rate proportional to its size. Write $P (t)$ . $P (t) = 400 \cdot 2^{t / 2}$
$\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (10) = 150$ . Find $k$ exactly. $k = - \frac{\ln 4}{10}$
A quantity follows $\frac{d y}{d t} = - 0.06 y$ . Find its half-life, to the nearest tenth. $11.6$
Solve $\frac{d y}{d t} = - 0.08 y$ , $y (1) = 250$ . $y (t) = 250 e^{- 0.08 (t - 1)}$
$\frac{d y}{d t} = k y$ , with $y (0) = 100$ and $y (3) = 125$ . Find $k$ exactly. $k = \frac{1}{3} \ln ⁣ (\frac{5}{4})$
A substance has a half-life of $3$ years. Starting from $1000$ g, how much is left after $10$ years? Round to the nearest tenth. $99.2$ g
A substance has a half-life of $6$ years and $N (0) = 1200$ . It changes at a rate proportional to its size. Write $N (t)$ . $N (t) = 1200 \cdot {(\frac{1}{2})}^{t / 6}$
$\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (5) = 450$ . Find $k$ exactly. $k = \frac{1}{5} \ln ⁣ (\frac{3}{2})$
A quantity follows $\frac{d y}{d t} = 0.1 y$ . Find its doubling time, to the nearest tenth. $6.9$
Solve $\frac{d y}{d t} = 0.04 y$ , $y (3) = 50$ . $y (t) = 50 e^{0.04 (t - 3)}$
$\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (2) = 50$ . Find $k$ exactly. $k = - \frac{\ln 4}{2}$
A substance has a half-life of $3$ years. Starting from $1200$ g, how much is left after $32$ years? Round to the nearest tenth. $0.7$ g
A population doubles every $3$ years and $P (0) = 150$ . It changes at a rate proportional to its size. Write $P (t)$ . $P (t) = 150 \cdot 2^{t / 3}$
$\frac{d y}{d t} = k y$ , with $y (0) = 300$ and $y (7) = 750$ . Find $k$ exactly. $k = \frac{1}{7} \ln ⁣ (\frac{5}{2})$
A substance has a half-life of $10$ years. Starting from $1500$ g, how much is left after $37$ years? Round to the nearest tenth. $115.4$ g
A substance has a half-life of $10$ years and $N (0) = 600$ . It changes at a rate proportional to its size. Write $N (t)$ . $N (t) = 600 \cdot {(\frac{1}{2})}^{t / 10}$
$\frac{d y}{d t} = k y$ , with $y (0) = 50$ and $y (5) = 40$ . Find $k$ exactly. $k = - \frac{1}{5} \ln ⁣ (\frac{5}{4})$
A substance has a half-life of $3$ years. Starting from $50$ g, how much is left after $35$ years? Round to the nearest tenth. $0.0$ g
Solve $\frac{d y}{d t} = 0.02 y$ , $y (4) = 1200$ . $y (t) = 1200 e^{0.02 (t - 4)}$
$\frac{d y}{d t} = k y$ , with $y (0) = 600$ and $y (2) = 1500$ . Find $k$ exactly. $k = \frac{1}{2} \ln ⁣ (\frac{5}{2})$

How to do these

$\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (4) = 600$ . Find $k$ exactly.

The solution is y = 200e^(kt).
At t = 4: 600 = 200e^(4k), so e^(4k) = 3.
Take logs: 4k = ln 3, so k = (ln 3)/4.

The answer is $k = \frac{\ln 3}{4}$ .

Where students go wrong

Using the percent as k. Growth of 5% a year compounded once is y = y(0)(1.05)^t, whose k is ln 1.05, not 0.05; k = 0.05 means continuous growth.

Also called dy/dt = ky, exponential models with differential equations, law of natural growth or continuous growth and decay.

What you can put on this worksheet

Solve with the initial value: Solve $\frac{d A}{d t} = - 0.15 A$ , $A (0) = 80$ . → $A (t) = 80 e^{- 0.15 t}$
Find k from two values: $\frac{d y}{d t} = k y$ , with $y (0) = 200$ and $y (7) = 1000$ . Find $k$ exactly. → $k = \frac{\ln 5}{7}$
Amounts, half-lives and times, rounded: $\frac{d P}{d t} = - 0.15 P$ and $P (0) = 400$ . Find $P (3)$ , to the nearest whole number. → $255$

Questions about these worksheets

Yes. Every exponential growth and decay (dy/dt = ky) sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the exponential growth and decay (dy/dt = ky) questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 3: solve with the initial value, find k from two values and amounts, half-lives and times, rounded. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the percent as k. Growth of 5% a year compounded once is y = y(0)(1.05)^t, whose k is ln 1.05, not 0.05; k = 0.05 means continuous growth.

Exponential growth and decay (dy/dt = ky) worksheet

Select your difficulty

Growth and Decay: dy/dt = ky

Growth and Decay: dy/dt = kyAnswer keyVersion 1

How to do these

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