Year 10 Exponential graphs and equations — Hard exercise 10

Wide tables scroll horizontally. For detailed working, you may prefer a larger screen.

Required skills (click to reveal)

A community garden in Adelaide uses an exponential model to track the growth of a native plant species introduced to restore soil health. The number of plants, P(t), after t weeks is given by P(t) = 92 × 3^(t/61).

After how many weeks will the number of plants reach exactly 276? Give your answer as an exact value in terms of logarithms.

Concept illustrations

These are separate worked examples of related concepts, not a diagram of this question. Their values and dimensions do not describe your question.

For y = 3 × 2ˣ, outputs 3, 6 and 12 occur at x = 0, 1 and 2. Equal horizontal steps multiply the output by 2.

x³ × x² has three x-factors followed by two more. That is five factors of x, so x³ × x² = x⁵.

Press Start when ready. Time includes time spent in another tab. Submit your answer before opening a help panel for a timed rank.

00:00

Report a problem or suggest an improvement

My notes — optional, not graded

What would you like to remember?0 words

Notes and scratch working stay in this browser and are linked to this question version. They are not graded or included in feedback. On shared devices, other people using this browser may see them.

Save as Include formulas and solutions

For this question only. Your full notebook is on the dashboard.

Notes save in this browser. Download a copy to keep them elsewhere.

Report a problem or suggest an improvement

Your observations help improve this question. You can report an issue or recommend a better explanation, visual or experience.

What would you like to share? Which area? What did you notice, or what would you improve?0 words

The report includes your message and question/version details. Saved profile details, answers, notes and attempt history are not automatically attached. Leave out names and contact details, and check screenshots for personal information and embedded metadata.

Optional screenshots (up to two PNG/JPEG files, 2 MiB each)

Attachments may contain personal information. Review them before sharing. Logs collect only event codes and timestamps after you enable them. Download the ZIP to keep a copy, then attach it to an email if you choose.

Formula and explanation (click to reveal)

Opening this before your first submission makes the attempt practice only.

Formula

Explanation

Worked solution and answer (click to reveal)

Opening this before your first submission makes the attempt practice only.

Worked solution

The problem gives the exponential growth model P(t) = 92 × 3^(t/61), where P(t) is the number of plants after t weeks. We are asked to find t when P(t) = 276.
Set up the equation: 92 × 3^(t/61) = 276.
Divide both sides by 92 to isolate the exponential term: 3^(t/61) = 276 / 92.
Calculate 276 ÷ 92: 276 ÷ 92 = 3.
Now the equation is 3^(t/61) = 3.
Since 3^1 = 3, the exponent must be 1. Therefore, t/61 = 1.
Multiply both sides by 61: t = 61.
To confirm, substitute t = 61 into the original equation: P(61) = 92 × 3^(61/61) = 92 × 3^1 = 92 × 3 = 276.
The solution satisfies the original equation exactly.
The question asks for the time in weeks when the number of plants reaches exactly 276. The answer is 61 weeks.

Reference answer: 61 weeks

Alternative methods (click to reveal)

Shortcuts are shown only when applicable. Check the conditions before using one.

Earlier attempts for this question

#	Date / time	Your answer	Result	Seconds	Help revealed	Rank

Import this question’s history from another computerHistory stays in this browser when storage is available. Export it to your shared Drive folder and import it on another computer.

Worksheet details

Naming pattern: Topic_Grade_Complexity_YYYYMMDD_HH24MISS.html

Validation compares your answer with the generated reference. The model's reference can be wrong. Time ranks use an estimated student solving time, not a measured benchmark or a comparison with other students. An incorrect answer receives no speed rank.

NavGanitMate privacy & saved work

Saved in this browser. Your profile preferences, answers, practice history and notes are stored here when browser storage is available. They are not automatically uploaded to our servers. People sharing this browser may see saved work; changing the learner name does not create a separate private account. A nickname is enough.

Keep copies of important work. Clearing browser data, private browsing or changing browser or device can mean losing saved work. Download notes and export history separately. Downloads are copies of the selected content, not a complete account backup; not every format can be imported to restore your work. Keep these files somewhere private.

Visiting the website. Cloudflare helps deliver and protect this site and processes connection information such as IP addresses and requested pages. Server error logs may also contain request details. Security features may use cookies. This is separate from your browser-saved learning records. Cloudflare privacy information.

A parent-connection reminder uses a local list of up to five completed question IDs for this browser profile and a tab-session dismissal marker. Reminder activity is not uploaded and practice remains available.

Sharing feedback is your choice. A report includes your message and question/version details. Your saved profile, answers and notes are not automatically attached. Files you select can reveal personal details or contain embedded metadata: check and redact them before sharing. Diagnostic event codes and timestamps are included only after you enable logging. Sending email shares your message and attachments with the recipient and email providers. A download alone sends nothing to us. Website feedback delivery may be unavailable; only a confirmed receipt indicates it was received.

Formula

Explanation

Worked solution

Earlier attempts for this question

Files ready to save