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### Course: Algebra 2>Unit 7

Lesson 3: Advanced interpretation of exponential models

# Interpreting change in exponential models: changing units

Sal analyzes the rate of change of various exponential models for different time units by manipulating the functions that model the situations.

## Want to join the conversation?

• I noticed that about half way through the video, when the table is filled out, the initial value of 315 was written as 3.5. I suspect this was a mistake and the value Sal meant to say was 315 the whole time?
• If you are watching in fullscreen, you can exit fullscreen and see that there is a correction box at the bottom right-hand corner that lets you know he messed up. Happens all the time. EDIT: Just realized this is 7 years late...
• Given the phrasing of the question, wouldn't the answer be 1.01? I'm looking to see if this question comes up in the quiz, and if we are marked wrong because of putting in too many decimal places. Sal didn't prep very well for this one.
• Yes. If you round the answer as requested by the instructions, 1.0058 would become 1.01.
• What I did was said that if in 10 years it went up 6%, then yearly that means it goes up 0.6%, which is close to the final answer of 0.58%. Was what I did acceptable?
• No, because it's not a linear function. The error in reasoning can be seen more clearly if we use a more exaggerated example. You can't say that if in one year you get 100 percent increase (or double the original), then in 10 years you'll get 1000 percent, or 10 times as much. Try doubling a number 10 times and see the difference.
• The first value of A(t) in the table is 315, as it goes on the value becomes 3.5. Is this a mistake?
• Yes, Sal slipped up. There’s usually a little box in the bottom right corner with corrections like that, though you can’t see the box if you’re in fullscreen.
(1 vote)
• Using this logic, can we solve compound interest problems involving fractional periods?
• Yes. You can manipulate the exponent t to however your situation is. For example, if it happens every year, you can put (t/365) for the exponent. Likewise, if it's every month, you can put (t/12) — however, be wary of how the question uses the variable.

Hopefully that helps !
• So, when I tried to model the function in terms of years instead of decades, intuitively I thought that it should look something like this:

a_years(t) = 315 * (1.06)^((1/10)t)

because in order for a_years to output an equivalent answer to a_decades, you should need to input 10 times more input into a_years. 1 year is a tenth of a decade, after all.

but Sal's model is slightly different
a_years(t) = 315 * (1.0058)^t

The first model is correct, right?
• Both models are correct, because
1.06^(0.1𝑡) = (1.06^0.1)^𝑡 ≈ 1.0058^𝑡
• Where did he get the 3.5?
• That was a mistake. He meant to write 315.
• in in the video, he didn't round to two decimal places. why?
• he later mentions it but doesn't correct it
• Khan's answer is wrong. Answer is 1.01 because that's 1.0058 rounded to 2dp.
• He stated that the whole answer is arguably correct. I see your reasoning in the blatant wording of the question, but for the sake of learning the topic itself I think it would be best to focus on the answer instead of the rounding.

So you may be right with this detail, but do you understand the whole point of the video?
(1 vote)
• A(t)=315(1.06)^t
A(t)=315(1.06)^(0.1t/0.1)
A(t)=315(1.06)^(0.1t x (1/0.1))
A(t)=315(1.06^(1/0.1))^0.1t
A(t)=315(1.06^10)^0.1t

why doesn't this work, because i thought this is showing the increase for every 0.1t i.e every year?
• Starting with the original expression: A(t) = 315 * (1.06)^t

A(t) = 315 * (1.06)^(0.1t/0.1) - This step is correct, you are simply dividing the exponent 0.1t by 0.1, which is the same as multiplying it by 1.

A(t) = 315 * (1.06)^(0.1t x (1/0.1)) - This step is still correct, as multiplying by (1/0.1) is equivalent to dividing by 0.1.

A(t) = 315 * (1.06^(1/0.1))^0.1t - Here's where the issue occurs. You've applied the exponent rule (a^(bc) = (a^b)^c) to move the 0.1t from the base to the exponent. However, this rule only applies when the exponent is a constant. In your case, 0.1t is not a constant; it depends on the value of t. So, you cannot directly move it to the exponent.

A(t) = 315 * (1.06^10)^0.1t - Following from the incorrect step above, you end up with this expression, which isn't the correct representation.

To show the increase every 0.1t (every year), you need to express the growth factor (1.06) correctly in terms of the yearly growth. The correct expression would be:

A(t) = 315 * (1.06^(0.1t))

In this expression, (0.1t) represents the number of years, and raising 1.06 to the power of (0.1t) gives you the factor by which the population increases every year. This way, you are correctly accounting for the yearly growth rate and its cumulative effect over time.

Now, the expression A(t) = 315 * (1.06^(0.1t)) represents the increase in the population every 0.1t (every year) based on a yearly growth rate of 6%.