SME
Grade 11 Math · Problem type
Continuous growth
Use this for “$2,000 grows at 5% a year compounded continuously. How long until it reaches $3,000?”
Chart
y-intercept (0, 2,000)
A(8.109) = 3,000
This is an example. Type your own values to start a new problem.
A
=
P
e
·
Formulas
A
=
P
×
e
r
×
t
3,000
=
2,000
×
e
0.05 × 8.1093
Assumptions
•
e ≈ 2.71828 is what compounding more and more often approaches.
•
The rate r is a decimal: 5% is 0.05.
•
To find t, divide by P and take ln of both sides: rt = ln(A ÷ P).
Step-by-step
WE KNOW
A = $3,000, P = $2,000, r = 0.05
FIND
time (years) (t)
Step 1 · Find time (years): t
Use A = P × er × t
Divide
by
P
,
take
ln
of
both
sides,
then
divide
by
r
.
t
=
ln(
A
÷
P
)
r
t
=
ln(3,000 ÷ 2,000)
0.05
e
0.05
t
=
1.5
0.05
t
=
ln
1.5
t
=
0.4055
0.05
t
=
8.1093
Check
Put the numbers back in. Both sides should match.
3,000
≈
2,000
×
e
0.05 × 8.1093
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.