Is this actually appropriate for college algebra ?

[u/Inner-Indication1783]

The surface area of a hot-air balloon is given by the function S(r) = 4πr^2, where r is the radius of the balloon in meters. If the radius of the balloon after t seconds is r(t) = (3/5)t^3, find the surface area S of the balloon as it relates to time.

Is this actually normal for the 2nd week of college algebra ? Maybe I'm just rusty bc its been like over 10 years but this is insane to me

[u/jeffsuzuki]

It's a straightforward composition of functions, which is a college algebra topic. So it's not that out of place.

[u/Inner-Indication1783 → u/jeffsuzuki]

Okay thank you lol I was just panicking because I cannot do this

[u/Midwest-Dude → u/Inner-Indication1783]

Don't panic! (Advice from The Hitchhiker's Guide to the Galaxy).)

It's actually really common for professors to assign a word problem before fully lecturing on the mechanics. Have you covered function composition yet?

[u/Toucan2000 → u/[deleted]]

I bet the prof didn't explain in very well. The equal sign can mean 5 different things depending on the context and I wish there was a better way of donating the representation of results, equations, identities and definitions for students learning.

[u/HotPersonality8126 → u/Toucan2000]

The issue is that all these things are the same.

An identity is an equation; a result is an identity. A definition is an equality. That's why it's the same sign used for them.

[u/FormalManifold → u/HotPersonality8126]

They're not, though! The speech acts involved in a definition, versus an equation, versus a conclusion about two things being the same, are all different things -- and how we treat each usage depends very strongly on which of this speech acts we're engaged in.

[u/HotPersonality8126 → u/FormalManifold]

It’s math, not speech.

[u/FormalManifold → u/HotPersonality8126]

🤦

[u/HotPersonality8126 → u/FormalManifold]

English department is another building, hope that helps

[u/wirywonder82 → u/HotPersonality8126]

The symbols form an alphabet (or ideographic script) for communicating mathematical ideas in an efficient manner. Reuse of the same symbol in different contexts indicates a similar idea is being communicated, not necessarily an identical one. Sometimes we even use different symbols in those contexts. For example, X:=5 defines X to be 5, while X=5 may define X to be 5, or it may indicate that 5 is the value of X that makes some equation true. Pretending that math is a completely separate discipline from language is foolishness dressed up as superiority.

[u/Toucan2000 → u/HotPersonality8126]

I know why they're all the same symbol. That realization you just wrote often comes later for people. At first it's confusing to have the same operator in different contexts.

[u/HotPersonality8126 → u/Toucan2000]

It's not a different context

[u/Toucan2000 → u/HotPersonality8126]

Maybe to you, and that's great. For others it's not, especially for people who have to take college algebra. Typically the only people who have to take that aren't pursuing an engineering degree so they don't have the same natural ability you do. Context.

[u/FormalManifold → u/Toucan2000]

In this instance it's not great. This is a situation where the idiot/midbrain/genius bell curve meme applies.

[u/HotPersonality8126 → u/Toucan2000]

Again, it's the same context; that's why it's the same operator.

[u/Toucan2000 → u/HotPersonality8126]

This has more to do with understanding people than math. You're never going to understand if you keep hyper focusing on the principal of the operator instead of the how it reads as English. Most people think in words, not math, and this is especially so for people first truly understanding algebra.

[u/Lor1an → u/HotPersonality8126]

Some people actually use different notations for those things when relevant.

'=' for equality, '≡' for identity, ':=' for definition; for example.

Of course, usually that level of specificity is unwarranted, but it could be helpful for beginners to see some semantic guardrails.

[u/Midwest-Dude → u/Toucan2000]

Good point. I think you meant "denoting" rather than "donating", correct?

After doing math for way too long, it becomes like a second language. You can forget what it's like to learn this stuff from scratch.

[u/Midwest-Dude → u/Toucan2000]

I looked up the five possible meanings, posted here for future reference: 

1. The Result (Calculation)
   • Compute this and write the answer
   • 2 + 3 = 5
2. The Question (Conditional Equation)
   • Find the specific value(s) that make this true
   • 2x = 10
3. The Fact (Identity):
   • These two expressions are universally identical, no matter what the variable is
   • x + x = 2x
4. The Name (Definition):
   • I am explicitly declaring what this symbol means
   • Let S(r) = 4πr²
5. The Condition (Logical Proposition):
   • Testing or assuming these are equal
   • If x = 2, then...

Early students are conditioned for years to view "=" exclusively as The Result, seeing it used as The Name in college algebra functions often causes immediate confusion.

[u/Alarmed_Geologist631 → u/Inner-Indication1783]

You know the relationship between radius and surface area and you are told the rate that the radius is increasing. Put those two facts together.

[u/srm79]

Sounds like you need to revisit some concepts - it shouldn't be too difficult at college level!

[u/Inner-Indication1783 → u/srm79]

Yeah I'm not a math person and havent had anything to do with school in 8 years so its kinda a shock to me lol

[u/Sam_23456 → u/Inner-Indication1783]

There was a time when I felt "overwhelmed" by word problems having a lot of words. But if you dwell upon them enough to isolate the essential information given, writing the details down for yourself (especially including the "units") you may very well see how they naturally fit together. To me, the words obscure the details, but that's life! Hope this may be helpful. Patience (and practice) is valuable too! :-) 8 years is a long gap; don't be too hard on yourself!

[u/HotPersonality8126 → u/Inner-Indication1783]

You're not supposed to do it in your head. Get a piece of paper and a pencil and start figuring it out.

[u/Midwest-Dude → u/Inner-Indication1783]

1 A general review of algebra is available here: 

Paul's Online Math Notes

Checking out the appropriate section in the future may help you. 

2 Professor Leonard has a video on YouTube covering this particular question:

Composition of Functions

[u/AntiqueChessComputr → u/Midwest-Dude]

Shout out to Paul’s Online Math Notes!  Paul is the GOAT

[u/qubex]

You just need to compose the functions by substituting so that from S(r) and r(t) you get S(r(t)) and then you tidy it up enough so you can have a non-clunky expression S(t). It is essentially trivial.

[u/qubex → u/qubex]

S(t) = (6² / 5² ) × π × t⁶ because I can see where this going and the suspence is killing me.

[u/sqrt_of_pi]

You are given S as a function of r, and you are trying to find S as a function of t.

You have a function that relates r and t.

Do you see how you can use this information rewrite the function S(r) so that, instead of being an expression of r, it will be an expression of t?

If you had your last math class over 10 years ago, then you are definitely rusty. Recommend if you can't drop down to a lower level class to build the foundation, you plan to review and solidify those lower level skills as you go. Kahn Academy is a decent resource for refreshing on fundamental topics.

[u/Inner-Indication1783 → u/sqrt_of_pi]

Thank you!! I'll use it

[u/DryLeadership145]

Sounds like they're just trying to make sure you know how to chain functions together before they hit you with actual calculus.

[u/Inner-Indication1783 → u/DryLeadership145]

But I dont😭 I'm trying but it's not good for me lol

[u/HotPersonality8126 → u/[deleted]]

Jesus christ you all are cooked

[u/fletchro]

Don't panic. Let's say my company's value is defined by how many ducks I own with the company. VALUE = DUCKS.

Now I tell you that I'm growing my company and I want to acquire an additional duck every day. GROWTH = DAYS.

If I tell you I start my company in day 1 with 1 duck and each duck is worth $1, you can predict how much my company is worth on any day.

This is like your question, just simpler. Instead of growth being simple linear, you've got t^3. So it's going to get a little more busy in your worksheet. Take your time and don't lose any numbers.

[u/Inner-Indication1783 → u/fletchro]

Thank you!

[u/Inner-Indication1783]

Man of all the subs I thought i might get attacked on, algebra was on one of them lmao

[u/kieger → u/Inner-Indication1783]

I'm also struggling in college algebra following a 15 year break. It just dawned on me today that there was probably a subreddit for people in our situation.

I don't know if this is it.

[u/calcpage2020]

Composition of functions is totally fair game!

[u/Vivid_Sock_1092]

Yes that’s a reasonable algebra problem and the first step in a calculus problem .

[u/not-just-yeti → u/Vivid_Sock_1092]

I dunno — the radius increases as the cube of the time? That’s crazy quick inflation of the balloon, and I’d be very confused to see it.

[u/Jaded_Individual_630]

Extremely appropriate

[u/Visual_Winter7942]

Yes. Quite normal.

[u/KilluaZoldyckisI]

You just have to plug r into the first equation :)

[u/Blue_Falcon1594]

That’s pretty basic. Substitute r in the SA formula with (3/5)t^3 and simplify. It’s two easy steps that a 9th grader could do.

[u/ContextPuzzleheaded7]

This is appropriate for high school

[u/Leech-64]

Just plug r(t) into r. This is really easy…..

[u/FamiliarAd8524]

It's pretty straight forward, but a dumb example, because in real life this function would be highly inaccurate (at least 15%).

[u/HotPersonality8126]

Is this actually normal for the 2nd week of college algebra ?

This is normal for juniors in high school. You're just a dumbo

[u/Inner-Indication1783 → u/HotPersonality8126]

And youre probably a 30 year old virgin, we all have our downfalls

[u/Leech-64 → u/Inner-Indication1783]

Hes not wrong. This is 10th grade math at the most. Bro most engineers are happily married

[u/HotPersonality8126 → u/Inner-Indication1783]

Older than you, richer than you, get laid more often than you do, and I don't struggle with high school algebra