Year 11 Standard · Unit 5 Ready for the Diploma · Lesson 3 of 6

How do you find the time when the unknown is in the exponent?

The time you want sits in the power, so dividing does not free it. A table of values finds the first whole period by hand, and a graph on a GDC finds the exact crossing and checks it.

FunctionsGDC optional

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 1: in y = 5 × 2x, find y when x = 3.
    Answer40. 5 × 8.
  2. From Lesson 2: a value rises by 15% each year. What is b?
    Answer1.15.
  3. From Year 8: work out 1.5 × 1.5 × 1.5.
    Answer3.375. 1.5 × 1.5 = 2.25, then × 1.5.
Words used in this lesson
target
a value that you want a quantity to reach. For example: a stock of 500 items.
table of values
a list of t and y for t = 0, 1, 2, and so on. For example: the rows of a model by hand.
crossing
the point where the curve meets the target line. For example: the time at which the curve reaches the line.
first whole period
the smallest whole t at which the value reaches the target. For example: the first week with at least the target.
intersection
a point where two graphs meet. For example: found with a GDC tool.
doubling time
the time for a quantity to become twice as big. For example: the years for savings to double.

Waiting for a thousand

A reading app for schools in Lagos starts with 30 users. Its team expects the number of users to grow by 40% every week. The company has promised its funders that the app will pass 1000 users, and the team want to say how many weeks that will take.

Predict first: will it take fewer than 10 weeks, or more? Then think about why you cannot just divide 1000 by 30 and be done. Use the first figure to see how a target line meets a curve, and the second figure to see how a doubling year depends on the rate.

1.5
200
4first whole period at or above the target
3.4exact crossing time

Start 50, base 1.5, target 200: after 3 periods the value is 168.75, after 4 it is 253.125. The first whole period is 4; the curve crosses at about 3.4.

Show me the steps
  1. Start 50, base 1.5, target 200: the values are 50, 75, 112.5, 168.75 and then 253.125. The first whole period at or above 200 is 4, and the curve crosses at about 3.4.
  2. Raise the target to 500: the values go on 379.7 at period 5, 569.5 at period 6. The first whole period is 6 and the crossing is at about 5.7.
  3. Set the base to 2.0 with the target 500: the values are 50, 100, 200, 400, 800. The first whole period is 4 and the crossing is at about 3.3. A bigger base reaches the same target sooner.

Find the time by a table of values, then check with a graph.

  • The time is in the exponent, so you cannot divide it out. Write the values for t = 0, 1, 2, … and look for where they pass the target.
  • The first whole period is the first t whose value is at or beyond the target. For growth it is the exact crossing rounded up (the crossing itself if it is a whole number).
  • For a rise of r% a year, the time to double is about 70 ÷ r. That is a quick estimate, not an exact answer.
  • A graph gives the exact crossing as a decimal t. Read the question: if it asks for the first whole period, round up, not to the nearest.

Always check by putting the answer back into the model: the value at the answer must be on the correct side of the target.

5
15first whole year at 2 or more
14estimate: 70 divided by the rate

Growth 5% a year: after 14 years the value is 1.98, after 15 it is 2.08. It first reaches 2 in year 15; 70 ÷ 5 = 14.

Show me the steps
  1. Growth 5% a year: the value is 1.98 after 14 years and 2.08 after 15, so it first reaches 2 in year 15. The estimate 70 ÷ 5 = 14 is one year short.
  2. Slide to 10%: the value is 1.95 after 7 years and 2.14 after 8, so year 8. The estimate is 70 ÷ 10 = 7, again one year short, because the rule gives the exact crossing and the question asks for a whole year.
  3. Slide to 30%: the value is 1.69 after 2 years and 2.2 after 3, so year 3. The estimate 70 ÷ 30 is about 2.3. It is a rough rule, not an answer.

Do not round to the nearest whole number. If the curve crosses 150 at t = 3.2, the value at t = 3 is still below 150, so the first whole period is 4, not 3. And a decay question that says “below” is the same: the value must have gone beyond the target, so round the crossing up.

Worked examples

A value is y = 20 × 1.6t. Find the first whole t for which y is more than 100.

  1. Make a table: t = 0 gives 20, then 32, 51.2, 81.92 and 131.072.
  2. 81.92 is below 100 and 131.072 is above it, so the first whole t is 4.
  3. Check by the model: 20 × 1.64 = 131.072, which is more than 100.

A value is y = 900 × 0.6t. Find the first whole t for which y is less than 100, and compare with the GDC crossing.

  1. The values are 900, 540, 324, 194.4, 116.64, 69.98.
  2. 116.64 is above 100 and 69.98 is below, so the first whole t is 5.
  3. The GDC crossing is at t = 4.3 (to 1 decimal place). Rounding that to the nearest whole number gives 4, but at t = 4 the value is still 116.64, so the first whole t is 5.

Your go. The first step is given. A value is y = 18 × 1.5t. Find the first whole t for which y is at least 100.

  1. Start the table: t = 0 gives 18, and t = 1 gives 18 × 1.5 = 27.
  2. Keep multiplying by 1.5 until the value is 100 or more.
Check the answer

The values are 18, 27, 40.5, 60.75, 91.125, 136.6875. Period 4 gives 91.125, which is below 100, and period 5 gives 136.6875, so the first whole t is 5. A common slip is to stop at period 4 because 91 is close to 100.

With a GDC (optional)

Graph the model in one function slot and the target as a horizontal line in another. Choose a window that shows the crossing: for growth over about 15 periods, x from 0 to 15 and y from 0 to a little above the target. Then use the intersection tool (often under a “calc” or “graph” menu). A table of values with a step of 1 gives the same answer for whole periods.

The GDC gives a decimal time. Whether the question wants the exact time, or the first whole period, is for you to decide from the words. Do the table by hand first; everything here can be done by hand.

Check before you go on

1. A model gives y = 3, 6, 12, 24, 48, 96, 192 for t = 0, 1, 2, 3, 4, 5, 6. What is the first whole t for which y is at least 100?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. A model is y = 5 × 3t. What is the smallest whole t for which y is more than 1000?

3. A model is y = 800 × 0.5t. What is the smallest whole t for which y is less than 10?

Level 2 · Building

4. A quantity grows by 12% a year. What is the first whole year in which it is more than double its start?

5. A GDC shows that y = 40 × 1.3t reaches 200 at t = 6.1, to one decimal place. What is the first whole t for which y is at least 200?

Level 3 · Stretch

6. A value is y = 1200 × 0.85t. In which whole year does it first fall below half of its start, which is 600?

7. Plan P starts with R5 and doubles every week. Plan Q starts with R50 and adds R25 every week. In which whole week is P first worth more than Q?

Something to think about

The rule 70 divided by the rate gave 14 for a 5% rise, but the first whole year was 15. Why is the first whole year always at least as big as the exact crossing, and could the rule ever be too large?

One way to think about it

The curve is rising, so the first whole year at or above the target is the next whole number at or after the crossing. The rule estimates the crossing time itself, so it can be above or below the whole year. For 5% the crossing is about 14.2, and the rule gives 14. It is a quick approximation, not a bound.

Exit ticket

8. A GDC shows that y = 1000 × 0.7t meets the line y = 300 at t = 3.4, to one decimal place. What is the first whole t for which y is below 300, and why?