SME
Grade 12 Science · Problem type
How long a reserve lasts when use grows
Use this for “600 billion barrels are used at 15 billion a year, and use grows steadily by 2% a year. How many years will they last?”
Diagram
Each slice is one year’s use, starting at 15 billion barrels and growing 2% a year, so each is larger than the last.
The reserve runs out after 29.4 years, against 40 if use stays the same.
This is an example. Type your own values to start a new problem.
T (years)
=
ln
1
+
×
Q (billion barrels)
r (billion barrels a year)
÷
k
g
Growth in use each year
example · %
y
Years it lasts at this year’s use
worked out (not typed) · years
Formulas
k
=
g
100
0.02
=
2
100
T
=
ln(1 +
k
×
Q
÷
r
)
k
29.4
=
ln(1 + 0.02 × 600 ÷ 15)
0.02
y
=
Q
r
40
=
600
15
Assumptions
•
Use grows steadily (continuously) by g% a year: after t years it is r × eᵏᵗ, with k = g ÷ 100.
•
The yearly use adds up to Q = r × (eᵏᵀ − 1) ÷ k after T years; solving for T gives the formula.
•
No new reserves are found; y = Q ÷ r is how long it would last if use stayed at r.
Step-by-step
WE KNOW
Q = 600 billion barrels, r = 15 billion barrels a year, g = 2%
FIND
growth rate as a decimal (k), years it lasts as use grows (T), years it lasts at this year’s use (y)
Step 1 · Find growth rate as a decimal (k)
Use k = g ÷ 100
A percent is a number of hundredths.
k
=
2
100
k
=
0.02
Step 2 · Find years it lasts as use grows (T)
Use T = ln(1 + k × Q ÷ r) ÷ k
The
growing
use
adds
up
to
Q
when
eᵏᵀ
=
1
+
kQ
r
;
take
ln
of
both
sides
and
divide
by
k
.
T
=
ln(1 + 0.02 × 600 ÷ 15)
0.02
T
=
ln(1.8)
0.02
T
=
0.5878
0.02
T
=
29.4
years
Step 3 · Find years it lasts at this year’s use (y)
Use y = Q ÷ r
For
comparison:
how
many
years
the
reserve
lasts
if
use
stays
at
r
.
y
=
600
15
y
=
40
years
Check
Put the numbers back in. Both sides should match.
0.02
=
2
100
✓
29.4
≈
ln(1 + 0.02 × 600 ÷ 15)
0.02
✓
40
=
600
15
✓
Topic check
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