Please help.
f(t)=(4+√t)(4t^2-5)
This is what I have so far...
(4+√t)(8t)+(4t^2-5) (1/2t^-1/2)
Am struggling to get this in factored form. Am working on a multiple choice take home test and nothing I am getting matches.
the derivative you got is correct.
by distributing, we can further simplify this into:
32t + 8t^(3/2) + 2t^(3/2) - (5/2)t^(-1/2)
10t^(3/2) + 32t - (5/2)t^(-1/2)
hope this helps~ :)
sofar ok
why don't we expand this ... and simplify to get
32t + 10t^(3/2) - (5/2)t^(-1/2)
= 32t + 10t√t - 5/(2√t)
Since I don't know what your choices are, I don't know in which direction to go from here.
Here is a good trick to see if your answer is the same as theirs.
Pick any point on the original
e.g. let t = 1
f(1) = 5(4-5) = -5
so (1,-5)
so sub in t = 1 into our derivative answer to get
32(1) + 10(1)(1) - 5/(2(1)) = 39.5
Any of the choices not giving you 39.5 would be incorrect.
Here are my options
a) 10t^2+32t√t - 5 / √t
b) 20t^2-64t√t - 5 / 2√t
c) 20t^2+ 32t√t - 5 / 2√t
d) 20t^2+64t√t - 5 / 2√t
e) 10t^2-32t√t - 5 / √t
added
If c) is written as
(20t^2 + 32t√t - 5)/(2√t) it would be same as my answer.
The formed a common denominator
the way you typed it, only the -5 would be divided by 2√t
Thanks. I see my mistake now.
To factorize the expression f(t) = (4 + √t)(4t^2 - 5), we can follow these steps:
Step 1: Simplify the expression inside the parentheses.
(4 + √t)(4t^2 - 5)
Step 2: Distribute the first term (4) across the parentheses.
4 * 4t^2 - 4 * 5 + √t * 4t^2 - √t * 5
16t^2 - 20 + 4t^2√t - 5√t
Step 3: Combine like terms.
(16t^2 + 4t^2√t) - 5√t - 20
Step 4: Factor out common terms, if possible.
In this case, we can factor out t^2 from the first term and √t from the remaining terms.
t^2(16 + 4√t) - (√t + 20)
So, the factored form of f(t) is:
f(t) = t^2(16 + 4√t) - (√t + 20)
Now, double-check the answer choices in your multiple-choice test and see if this matches any of them.