MATHEMATICS COLLEGE

When we expand (2x + 1/2)^6, what is the coefficient on the x^4 term?

Answers

Answer 1

Answer:

Answer: The coefficient before x^4 is 60

Step-by-step explanation:

Hey! So I am not an expert at this, but you have to use the binomial theorem

I have attached of the Pascals Triangle (one shows the row numbering as well)

Basically in a pascal triangle, you add the two numbers above it to get the next number below

As you can see, the rows start from 0 instead of 1

The 6th row contains the numbers 1, 6, 15, 20, 15, 6, 1 which would be the coefficient terms

NOTE: the exponents always add to 6, the first term starts at 6 and decrease it's exponent by 1 each time (6, 5, 4, 3, 2, 1, 0) and the second term increases it's exponent by 1 each time (0, 1, 2, 3, 4, 5, 6)

Using this information the third term from the sixth row (15) would be where it is x^4 (I have circled it on the second image)

It would be 15 × 2^4 × (1/2)^2 = 60

The reason why it is 2^4 and (1/2)^2 is because the third term has the exponents 4 and 2 (bolded on the NOTE) which means that the first term must be put to the power of 4 and the second term must be put to the 2nd power

Sorry for the lousy explanation. I really hope this makes sense! Let me know if this helped :)

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Answers

Answer:8

Step-by-step explanation:

A bacterial culture starts with 500 bacteria and doubles in size every half-hour.(a) How many bacteria are there after 3 hours?
Answer: Since 3 hours equals 6 half-hours, the culture will have doubled 6 times.
Therefore, there will be
500 · 2
6 = 32,000
bacteria.
(b) How many bacteria are there after t hours?
Answer: Since t hours is the same as 2t half-hours, the culture will have doubled 2t
times. Therefore, there will be
500 · 2
2t
bacteria.
(c) How many bacteria are there after 40 minutes?
Answer: There are two possible answers depending on how you interpret the set-up
to the problem. If each bacterium in the culture doubles once every half-hour on the
half-hour, then each one will double after exactly 30 minutes, and then not again until
60 minutes have passed. In that case, there will be
500 · 2 = 1000
4
bacteria after 40 minutes.
On the other hand, if each bacterium doubles exactly once per half-hour, but at some
random time within that half-hour, then it makes sense to think of the population
function P(t) = 500 · 2
2t as continuous. In that case, since 40 minutes is
40
60
=
2
3
of an hour, the population will be
500 · 2
2
2
3 = 500 · 2
4
3 ≈ 1259
after 40 minutes.

Answers

Answer:

a). 32000

b). $T_(t)=500* 4^(t)$

c). 1259

Step-by-step explanation:

Growth of a bacteria is always exponential. Therefore, population of the bacteria is represented by the the geometric sequence.

Sum of the bacterial population after t hours will be represented by

$T_(n)=ar^(n)$

Where a = population at the start

r = ratio with the population is growing

n = time or duration of the growth in one hour

a). Population of 500 bacteria gets doubled after half an hour.

Or gets 4 times after an hour

This sequence will have a common ratio r = 4

and initial population a = 500

Therefore, population of the bacteria after 3 hours will be

$T_(3)=500* 4^(3)$

$T_(3)=32000$

b). After t hours number of bacteria will be represented by

$T_(t)=500* 4^(t)$

c). We have to calculate the population after 40 minutes.

That means duration 't' = 40 minutes of $(2)/(3)$ hours

By the formula,

$T_{(2)/(3)}=500* 4^{(2)/(3)}$

$T_{(2)/(3)}=1259.92$ ≈ 1259

Therefore, number of bacteria after 40 minutes will be 1259.

After 3 hours, there will be 32,000 bacteria in the culture, given that the bacteria double in size every half-hour. The number of bacteria at any given time depends on whether they double precisely every half-hour or continuously within that timeframe.

In this scenario, the population growth of the bacterial culture follows exponential growth, where it doubles every half-hour. To calculate the number of bacteria after 3 hours (equivalent to 6 half-hours), you can use the formula for exponential growth: P(t) = P₀ * $2^{(t/h)$ , where P(t) is the population at time t, P₀ is the initial population, t is the time in hours, and h is the time interval for doubling (in this case, 0.5 hours). Plugging in the values, you get P(3) = 500 * $2^{(3/0.5)$ = 32,000 bacteria.

This means that after 3 hours, there will be 32,000 bacteria in the culture. The explanation also addresses the alternate interpretation of continuous growth, where the population increases continuously within each half-hour, resulting in approximately 1259 bacteria after 40 minutes.

Learn more about time here: brainly.com/question/34222581

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Answers

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What is the best estimate of the circumference of a circle with a radius of 8 inches

Answers

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(1) -3x-4y+11z from-9y+6z-3x
(2) 3x⁴-4x³+7x-2 from 9-7x⁴+6x³-2x²-11x

Answers

Answer:

1) 5y + 5z

2) 10x⁴ - 10x³ + 2x² + 18x - 11

Step-by-step explanation:

Given the subtraction of the following polynomial expressions:

(1) -3x - 4y + 11z from -9y + 6z - 3x

In order to make it easier for us to perform the required mathematical operations, we must first rearrange the terms in the subtrahend by alphabetical order.

-3x - 4y + 11z

-3x - 9y + 6z ⇒ This is the subtrahend.

Now, we can finally perform the subtraction on both trinomials:

$\displaystyle\mathsf{\left \ \quad\:\:\:\:{-3x - 4y + 11z} \atop -\quad{\underline{-3x - 9y + 6z\:\:\underline}} \right.}$

In the subtrahend, the coefficients of x and y are both negative. Thus, performing the subtraction operations on these coefficients transforms their sign into positive.

$\displaystyle\mathsf{\left \ \quad\:\:\:\:{-3x - 4y + 11z} \atop -\quad{\underline{-3x - 9y + 6z\:\:\underline}}\right.} \n\qquad\sf {\qquad\:\:\:0x\:+\:5y\:+5z$

The difference is: 5y + 5z.

(2) 3x⁴- 4x³ + 7x - 2 from 9 - 7x⁴ + 6x³- 2x² - 11x

Similar to the how we arranged the given trinomials in Question 1, we must rearrange the given polynomials in descending degree of terms before subtracting like terms.

3x⁴- 4x³ + 7x - 2 ⇒ Already in descending order (degree).

9 - 7x⁴ + 6x³- 2x² - 11x ⇒ -7x⁴ + 6x³- 2x² - 11x + 9

In subtracting polynomials, we can only subtract like terms, which are terms that have the same variables and exponents.

$\displaystyle\mathsf{\left \ \quad\:\:{3x^4\:-4x^3\:+\:0x^2\:+\:7x\:-\:2} \atop -\quad{\underline{-7x^4\:+6x^3\:-2x^2\:-11x\:+\:9 \:\:\underline}}\right.}$

In the minuend, I added the "0x²" to make it less-confusing for us to perform the subtraction operations.

The same rules apply in terms of coefficients with negative signs in the subtrahend, such as: -7x⁴, - 2x², and - 11x ⇒ their coefficients turn into positive when performing subtraction.

$\displaystyle\mathsf{\left \ \quad\:\:{3x^4\:-4x^3\:+\:0x^2\:+\:7x\:-\:2} \atop -\quad{\underline{-7x^4\:+6x^3\:-2x^2\:-11x\:+\:9 \:\:\underline}}\right.} \n\qquad\sf {\qquad\:\:10x^4-10x^3+2x^2+18x\:-11$

Therefore, the difference is: 10x⁴ - 10x³ + 2x² + 18x - 11.

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