Can anyone explain how to do mathematical induction. The question is below. Brainliest awarded for the right answer. Thanks.
Answers
The sum of the triangular numbers is [tex]\frac{n(n+1)(n+2)}{6}[/tex]
According to the question, the series of triangular numbers is given as in the form of the number of dots constituting the equilateral triangles i.e.
1, 3, 6, 10, 15 . . . . .n
triangular number are the sequence and series of the numbers and each number represents and constitute in visualization of series of equilateral triangle.
given series in the figure
1, 3, 6, 10, 15, . . . . . . ., n
each number represents the number of dots containing in triangle
now the series can be given by
[tex]S = \frac{n(n+1)}{2}[/tex] for n = 1, 2, 3, 4,. . . . .n
now , according to the question the sum of the series can be given as
⇒ ∑S
⇒ ∑[tex]\frac{n^2+n}{2}[/tex]
⇒ [tex]\frac{1}{2}[\sum n^2+\sum n]\\\frac{1}{2}[\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}] \\\frac{n(n+1)}{4}[\frac{(2n+1)}{3}+1] \\\frac{n(n+1)}{4}[\frac{(2n+1+3)}{3}] \\\frac{n(n+1)}{4}[\frac{(2n+4)}{3}] \\\frac{n(n+1)}{4}2[\frac{(n+2)}{3}]\\\frac{n(n+1)(n+2)}{6}[/tex]
Thus, the sum of the triangular number is given by [tex]\frac{n(n+1)(n+2)}{6}[/tex]
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Related Questions
The graph of a function f(x) is shown. What is the value of x where f(x) = -5? And f(0)?
Answers
Answer:
x = -3 when f(x) = -5
f(0) = 3
Step-by-step explanation:
When f(x) = -5, that means they have told you the y value. When y = -5, x = -3.
If the number is in the parentheses of the function, they are giving you the x value. When x = 0, y = 3
A roll of paper towels is wound around a hollow cardboard tube. the
cardboard tube in the middle of the roll has an outside radius of 2.4 cm. the
thickness of the paper is 0.3 mm. the sequence of distances of the loops of
paper away from the center of the roll (in centimeters) is the following:
21, 22, az, a4 a5, ... = 2.40, 2.43, 2.46, 2.49, 2.52, ..
what is the radius of the 85th loop of paper, starting from the center of the
tube?
a. 4.95 cm
b. 4.86 cm
c. 4.92 cm
d. 4.89 cm
Answers
The correct Option is Option C: The radius of the cardboard tube of 85th loop of paper is 4.92cm
The arithmetic sequence is the sequence where every term is increased or decreased by a fixed number from the previous number.
Here the outer radius of the tube is 2.4 cm
the thickness of the paper is 0.3mm= 0.03cm
i.e. in every loop the increase in the radius of the loop is 0.03cm
then the radius in every sequence will be 2.40, 2.43, 2.46, 2.49, 2.52, .....
so here it is clear that it is an arithmetic sequence with a common difference of 0.03.
nth term of the sequence, aₙ = a₁ + (n - 1)d where a₁ is the first term, n is the index of the loop, and d is a common difference.
here a₁ =2.40
d=0.03
n=85
the radius of tube of the 85th loop will be= r= 2.40+(85-1)0.03= 2.40+ 2.52= 4.92
Therefore The radius of the cardboard tube of the 85th loop of paper is 4.92cm
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answer both questions please
Answers
Answer: The two imaginary solutions with rational coefficients are +/- 4i and the two imaginary solutions with irrational coefficients are +/- 1.414213562i.
Step-by-step explanation:
I would recommend using pascal's triangle to solve this problem.
*Note: This is more of a quick answer, I can show a detailed step-by-step procedure if you need, just comment.
Sorry I only have the answer key, lol, but for #13, can someone explain to me where they got 4.6 from?? ty and I don't need an explanation for 14; I just couldn't crop the photo
Answers
The measure of the side RS from the figure is 4.6 units
Midpoint theorem of a trapezium
The given figures are trapezoid. Given the following parameters
PQ = 4RS
We need to determine how PR is equivalent to 4.6
Since PQ = 18.4 from the diagram, hence;
18.4 = 4RS
RS = 18.4/4
RS = 4.6 units
Hence the measure of the side RS from the figure is 4.6 units
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given triangle PQR with QR = 8, PQ = 6, and m p = 90 degrees, which of the following choices is used to determine m r
Answers
The value of the angle R in the right triangle PQR is sin⁻¹ 6 / 8
The situation forms a right angle triangle.
How to find an angle in a right triangle?The angle R can be found as follows:
using trigonometric ratios,
sin R = opposite / hypotenuse
Therefore,
opposite sides = 6
hypotenuse = 8
sin R = 6 / 8
R = sin⁻¹ 6 / 8
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An ongoing promotion at a department store gives customers 20\%20%20, percent off the portion of their bill that is over \$100$100dollar sign, 100. Ruby's total bill at the department store after the promotion has been applied is \$250$250dollar sign, 250. If xxx represents the amount of money Ruby would have spent on the same purchase at the department store without the promotion, which of the following equations best models the situation?
Answers
The equation best models the situation 100 + 0.8(x - 100) = 250.
The correct option is (D)
What is equation?
An equation in math is an equality relationship between two expressions written on both sides of the equal to sign.
The complete question is:
An ongoing promotion at a department store gives customers 20% off the portion of their bill that is over $100. Ruby's total bill at the department store after the promotion has been applied is $250. If x represents the amount of money Ruby would have spent on the same purchase at the department store without the promotion, which of the following equations best models the situation?
A. 0.2x + 100 = 250
B. 0.8x + 100 = 250
C. 100 + 0.2(x - 100) = 250
D. 100 + 0.8(x - 100) = 250
Given:
Department store gives 20 % off the portion of the bill that is over 100 dollars.
Total bill = 250
let x is the amount of money she would have spent without the promotion.
Then, cost after discount applies = 0.8 ( x - 100 ).
Hence, the equation is 100 + 0.8 (x - 100) = 250.
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Peter's garden is in the shape of a rectangle whose width is 15 inches and length is twice the width. Find the area of the garden.
Answers
Answer:
450 in^2
Step-by-step explanation:
Width = 15 Length is twice as much = 30 in
Area = L * W = 15 * 30 = 450 in^2
Which graph represents the solution of x² + y² < 25 and y² <6x?
Answers
The graph is shown below
What is graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
As, the equation x² + y² < 25 represents equation of circle.
So, the graph will be of the form circle.
and, y² <6x is the equation to represent parabola
So, the graph will represent a parabola.
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A school is watching students as they enter the football game for students who are dressed
inappropriately. they estimate that 7% of all students are dressed inappropriately. out of a group
of 40 students, what is the probability that exactly 2 are dressed inappropriately? round to 3 decimal places.
Answers
Using the binomial distribution, it is found that there is a 0.242 = 24.2% probability that exactly 2 are dressed inappropriately.
What is the binomial distribution formula?The formula is:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]
The parameters are:
x is the number of successes.n is the number of trials.p is the probability of a success on a single trial.The values of the parameters are given as follows:
n = 40, p = 0.07.
The probability that exactly 2 are dressed inappropriately is given by P(X = 2), hence:
[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]
[tex]P(X = 2) = C_{40,2}.(0.07)^{2}.(0.93)^{38} = 0.242[/tex]
0.242 = 24.2% probability that exactly 2 are dressed inappropriately.
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PLEASE HELP ASAP
The file is attached
Answers
The trigonometric function for an angle [tex]\dfrac{\pi}{4}[/tex] is
[tex]tan\theta =1\ \ \ \ Cos \theta=\dfrac{\sqrt{2}}{2}\ \ \ \ Sin\theta = \dfrac{\sqrt{2}}{2}[/tex]
The trigonometric function for an angle [tex]\dfrac{\pi}{6}[/tex] will be:-
[tex]tan\theta = \dfrac{1}{\sqrt3}}\ \ \ \ \ Sin\theta = \dfrac{1}{2} \ \ \ \ \ Cos \theta = \dfrac{\sqrt{3}}{2}[/tex]
What is trigonometry?The branch of mathematics sets up a relationship between the sides and the angles of the right-angle triangle termed trigonometry.
The trigonometric function for an angle [tex]\dfrac{\pi}{4}[/tex] is
[tex]tan\theta = tan\dfrac{\pi}{4}=1\ \ \ \ Cos \theta=Cos\dfrac{\pi}{4}=\dfrac{\sqrt{2}}{2}\ \ \ \ Sin\theta =Sin\dfrac{\pi}{4}= \dfrac{\sqrt{2}}{2}[/tex]
The trigonometric function for an angle [tex]\dfrac{\pi}{6}[/tex] will be:-
[tex]tan\theta=tan \dfrac{\pi}{6}= \dfrac{1}{\sqrt3}}\ \ \ \ \ Sin\theta =Sin\dfrac{\pi}{6}= \dfrac{1}{2} \ \ \ \ \ Cos \theta = Cos\dfrac{\pi}{6}= \dfrac{\sqrt{3}}{2}[/tex]
Therefore the trigonometric function for an angle [tex]\dfrac{\pi}{4}[/tex] is
[tex]tan\theta =1\ \ \ \ Cos \theta=\dfrac{\sqrt{2}}{2}\ \ \ \ Sin\theta = \dfrac{\sqrt{2}}{2}[/tex]
The trigonometric function for an angle [tex]\dfrac{\pi}{6}[/tex] will be:-
[tex]tan\theta = \dfrac{1}{\sqrt3}}\ \ \ \ \ Sin\theta = \dfrac{1}{2} \ \ \ \ \ Cos \theta = \dfrac{\sqrt{3}}{2}[/tex]
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[tex]2^{x^2-5x}=\frac{1}{64}[/tex]
Answers
Answer:
2, 3
Step-by-step explanation:
See attached images for work.
What is the mean absolute deviation of Patrick’s scores? Show your work
Answers
3 connecting lines are shown. Line D F is horizontal. Line D E is about half the length of line D F. Line F E is about one-third of the length of line D F.
Which inequality explains why these three segments cannot be used to construct a triangle?
EF + FD > DE
ED + EF < DF
ED + EF > DF
EF + FD < DE
Answers
Answer:
ED + EF < DF
Step-by-step explanation:
shorter side lengths added together > larger side length
however, smaller side lengths are < larger
So ED + ED < DF means that this cannot be a triangle
Answer:
ED + EF < DF
(second option listed)
Step-by-step explanation:
for a triangle, the sum of the two shorter side lengths must be greater than the larger side length
so we know that the two smaller lengths (DE and FE) would have to add up to be greater than the larger side length (DF)
because this is not the case, we know that we cannot construct a triangle
[to construct a triangle:]
smaller + smaller > bigger
DE + FE > DF; which is false/not the case
so because DE + FE < DF, we cannot form a triangle
so, ED + EF < DF explains why these three segments cannot construct a triangle
The first term of an arithmetic progression is 3 and the fifth term is 9. Find the number of terms in the progression if the last term is 81.
Answers
Answer:
n = 53
Step-by-step explanation:
the sum to n terms of an arithmetic progression is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
here a₁ = 3 and a₅ = 9 , then
3 + 4d = 9 ( subtract 3 from both sides )
4d = 6 ( divide both sides by 4 )
d = 1.5
then solving for n
3 + 1.5(n - 1) = 81 ( subtract 3 from both sides )
1.5(n - 1) = 78 ( divide both sides by 1.5 )
n - 1 = 52 ( add 1 to both sides )
n = 53
there are 53 terms in the progression
Answer:
53 terms
Step-by-step explanation:
Finding common difference :
a = 3a + 4d = 94d = 6d = 1.5Solving :
a_{n} = a + (n - 1)d81 = 3 + 1.5n - 1.579.5 = 1.5nn = 53The circle graph shows the results of a survey of registered voters the day of an election.
The error given in the graph means that the actual percentage could be 2% more or 2% less than the parent reported by the survey.
What are the minimum and maximum percentages of voters who could vote Republicans? Green?
How can you use absolute value equations to represent your answers in part (a)?
One candidate receives 44% of the vote, Which party does the candidate belong to? Explain.
Answers
The minimum and maximum percentages of voters who could vote Republicans are 40% and 44% respectively, while Green are 0% and 4%
The minimum and maximum percentages of voters in Republicans? Green?The attached circle graph represents the missing information in the question.
From the graph, we have:
Republicans = 42%
Green = 2%
The actual percentage is given as: ±2%
So, we have:
Republicans = 42% ± 2% = (40%, 44%)
Green = 2% ± 2% = (0%, 4%)
Hence, the minimum and maximum percentages of voters who could vote Republicans are 40% and 44% respectively, while Green are 0% and 4%
The absolute value equations of part (a)?We have:
The actual percentage is given as: ±2%
Let the number of votes be on the circle graph be x and the range be y.
So, we have:
y - x = ±2%
Express as absolute value equation
|y - x| = 2%
For Republicans, we have:
|y - 42%| = 2%
For Green, we have:
|y - 2%| = 2%
Hence, the absolute value equations are |y - 42%| = 2% and |y - 2%| = 2%
The party of candidate that receives 44% voteIn (a), we have:
Republicans = 42% ± 2% = (40%, 44%)
Hence, the party of candidate that receives 44% vote is Republican
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Jane and David had identical chocolate bars.
David ate 78 of his chocolate bar.
Jane ate more of her chocolate bar than David.
What fraction of her chocolate bar could Jane have eaten?
Answers
Answer:
Step-by-step explanation:
I'm assuming you meant "David at 78% of his chocolate bar" which means he at 78/100 of his chocolate bar. So if Jane ate more she could've eaten n/100 of her chocolate bar where [tex]78 < n \leq 100[/tex]. Greater than 78 since David at 78/100 and less than or equal to 100 since 100/100 means Jane at the entire thing at 101/100 wouldn't make any sense.
ignore the answer i clicked on, i did not mean to click it
Answers
Answer:
20√2Step-by-step explanation:
The diagonal of a square is given as, √2 × side. Rearranging this formula to calculate the side of the square, we get, side = diagonal/√2 = (√2 × diagonal)/2 . Thus, the perimeter of the square can be calculated using the formula, P = 2√2 × diagonal.
Diagonal = 5 * 2 = 10
P = 2√2 * 10
P = 20√2
You work for a lending institution and are tasked with whether or not to approve a home loan, using the standard 28/36 ratio. the loan application is for $230,000. you see that the applicant has an annual salary of $83,000. the applicant also has a car payment of $315, a student loan of $140 and a boat loan of $96. how likely are you to approve the loan? a. very likely; recurring debt is considerably less than what is allowed. b. somewhat likely; recurring debt is very close to what is allowed. c. not likely; recurring debt is higher than what is allowed. d. there is not enough information given to determine the answer.
Answers
You work for a lending institution and are tasked with whether or not to approve a home loan, its Somewhat likely; that recurring debt is very close to what is allowed. Option B.
What is recurring debt?Generally, any payment that is utilized to repay debt obligations that occur on a continuous basis is said to be recurrent debt.
In conclusion, Recurring debt consists of payments that cannot be readily canceled at the request of the payer. Examples of this kind of debt include alimony, child support, and loan installments.
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Answer:
its B
Step-by-step explanation:
100 POINTS!!!!! WILL GIVE BRAINLIEST!!!!!! PLS HELP ASAP!!!!!!!
SHOW ALL WORK!!!!!!! EXPLAIN ALL WORK ASWELL!!!!!!!!!
1.
A theater has 20 rows of seats. If there are 4 seats in the 1st row 12 in the 2nd row, 20 in the 3rd row . How many seats are there in total? Show and explain all work.
Answers
Answer:
1,600 seats in the theater
Step-by-step explanation:
well it seems like each row is being increased by 8 seats each time.
we start with 4 seats in the 1st row. 12 in the 2nd. and 20 in the 3rd. I will make a chart showing you how many seats are in each row.
4th row - 28 5th row - 36 6th row - 44 7th row - 52 8th row - 60 9th - 68 10th - 76 11th - 84 12th - 92 13th - 100 14th - 108 15th - 116 16th- 124 17th - 132 18th - 140 19th - 148 20th - 156
if we add these all together you will get 1,600
Answer:
1600 seats
Explanation:
1st row = 4 seats, 2nd row = 12 seats, 3rd row = 20 seats
This is arithmetic progression with first term (a) being 4 and common difference (d) of 8 and total term (n) of 20.
Sum of arithmetic series:
[tex]\sf S_n = \dfrac{n}{2} ( 2a + (n-1)d)[/tex]
insert values
[tex]\sf S_{20} = \dfrac{20}{2} ( 2(4) + (20-1)8)[/tex]
[tex]\sf S_{20} = 1600[/tex]
please help
It is Ms. Smith’s birthday, and her students want to surprise her with a room full of balloons. Use the Fermi process to estimate the number of balloons needed to fill Ms. Smith’s classroom. Assume the classroom is a rectangular prism with a length of 38 ft, a width of 52 ft, and a height of 12 ft. Assume the balloons are spheres with a radius of 0.35 ft. Show your work.
Answers
The number of balloons needed to fill the classroom are 46240 balloons (approx)
Concept: Fermi process is the technique that helps to formulate an answer to a problem based on a series of logical assumptions.
No. of balloons = Volume of rectangular prism/ Volume of each sphere (balloon)
Given: The classroom is a rectangular prism with a length of 38 ft, a width of 52 ft, and a height of 12 ft, and the balloons are spheres with a radius of 0.35 ft.
No. of balloons = Volume of rectangular prism/ Volume of each sphere (balloon)
No. of Balloons = 38*52*12/3.14*0.35*0.35*(4/3)
No. of Balloons = 23712/0.5128 = 46240 balloons (approx)
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Roni wants to write an equation to represent a proportional relationship that has a constant of proportionality equal to StartFraction 7 over 25 EndFraction. She writes the equation y = x + StartFraction 7 over 25 EndFraction. What error is Roni making?
She should have written y = negative x + StartFraction 7 over 25 EndFraction so that x and y have a constant sum.
She should have written x y = StartFraction 7 over 25 EndFraction so that x and y have a constant product.
She should have written y = StartFraction 7 over 25 EndFraction x so that x and y have a constant quotient.
She should have written y = StartFraction 7 over 25 EndFraction so that y has a constant value.
Answers
Answer:
y = 7/25x
Step-by-step explanation:
For a proportional relationship, we have
y ∞ x.
Let k = constant of proportionality = 7/25
Removing the proportionality sign, we have
y = k x
Therefore, the answer is
y = 7/25x
Instead of y= x + 7/25
So, the error is that x should be multiplied by 7/25 and not added.
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A man 1.8 m tall observes the angle of elevation of a tree to be 26 degrees, if he is standing bf 16 m from the tree find the height of the tree
Answers
The height of the tree is the man's height plus x.
[tex]\tan 26^{\circ}=\frac{x}{16}\\x = 16 \tan 26^{\circ}[/tex]
So, the height of the tree is [tex]\boxed{(16 \tan 26^{\circ}+1.8) \text{ m}}[/tex]
The number of gallons of gas Connie’s car, g, uses is directly proportional with the number of miles she drives, m. Last week, she drove 469.8 miles and used 14.5 gallons of gas. Which equation would best represent the problem?
Answers
The equation which represent the problem is 14.5 = 469.8k.
What is Equation?An equation is a mathematical statement with an 'equal to =' symbol between two expressions that have equal values.
Here, As given in question
The number of gallons of gas Connie’s car, g, uses is directly proportional with the number of miles she drives, m.
g α m
g = km ............(i)
g = 14.5 gallons of gas m = 469.8 miles
14.5 = k X 469.8
14.5 = 469.8k
Thus, the equation which represent the problem is 14.5 = 469.8k.
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Indigo earned a grade of 93% on her multiple choice history final that had a total of
200 problems. How many problems on the final exam did Indigo answer correctly?
Answers
Answer:
186
Step-by-step explanation:
x/200 = 0.93
x = 200(0.93)
x = 186
The number of problems that Indigo did answer correctly in the final exam is 186.
Given that:
Indigo earned a grade of 93% on her multiple-choice history final which had a total of 200 problems.
Total number of problems = 200
So, 100% grade = 200 problems
Let x be the number of problems which is answered correctly to get 93%.
Using the proportional concept,
[tex]\frac{x}{200} =\frac{93}{100}[/tex]
Cross multiply,
100x = 93 × 200
Dividing whole by 100,
x = 93 × 2
x = 186
Hence, the number of problems that are answered correctly is 186.
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allowing 16% discount on the MP of a television and leying 13% Vat buyer has pay RS 18984 to buy it . Find MP of that television
Answers
The marked price of the television is Rs. 20,000.
What is Marked Price?The market price is the current price at which a good or service can be purchased or sold.
Here, let the marked price of the television be Rs. x
Discount = 16% on MP
VAT = 13% additional on MP
Total paid amount = Rs. 18984
Now, according to question;
MP X (1 - Discount) X (1 + VAT) = 18984
x (1 - 0.16)(1+0.13) = 18984
x = 18984/(0.84 X 1.13)
x = 18984 / 0.9492
x = 20,000
Thus, the marked price of the television is Rs. 20,000.
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What was k? 200 400 600
Answers
Answer:
pls yo give the full question
Given cos theta = √3/3 and sin theta < 0. What is the value of sin theta?
Answers
The value of the sin theta will be equal to [tex]Sin\theta =\sqrt{\dfrac{2}{3}}[/tex]
What is trigonometry?The branch of mathematics sets up a relationship between the sides and the angles of the right-angle triangle termed trigonometry.
Given that:-
Cos[tex]\theta[/tex] = √3/3 Sin[tex]\theta[/tex] <0The value of Sin will be calculated as:-
[tex]Sin^2\theta +Cos^2\theta = 1\\\\Sin^2\theta + \dfrac{\sqrt{3}}{2}=1\\\\Sin^2\theta = 1 -\dfrac{1}{3}\\\\Sin\theta = \sqrt{\dfrac{2}{3}}[/tex]
We can see that the value of sin is less than zero therefore
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Solve the following quadratic by
completing the square.
y = x2 - 6x +2
Answers
Answer:
Roots of the equation are (−3−√7) , (−3+√7)
Step-by-step explanation:
For solving a quadratic equation using completing the square method:
set the value of 'y' = 0add and subtract the square of half of the coefficient of x.solve the obtained equation for 'x'given equation:
y = x^2 - 6x +2
0 = x^2 - 6x + 2
0 = x^2 - 3^2 + 3^2 -6x +2
0 = (x-3)^2 -7
7 = (x-3)^2
±[tex]\sqrt{7}[/tex] = x-3
±[tex]\sqrt{7}[/tex] + 3 = x
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A triangle has two sides of lengths 10 and 14 what value could the length of the third side be
Answers
Answer:
17.20465053
Step-by-step explanation:
17.20465053 or
17.2 (to 1 dp)
Answer:
Third side has to be greater than 4 and less than 24.
Step-by-step explanation:
A triangle is valid if the sum of two sides is greater than the third side. That has to be the case for each of the sides. So let the sides of triangle be a, b and c.
a + b > c
a + c > b
b + c > a
Given:
a = 10
b = 14
We are looking for the third side, c.
First inequality:
a + b > c
10 + 14 > c
24 > c
This tells us that c has to be less than 24.
Second inequality:
a + c > b
c > b - a
c > 14 - 10
c > 4
This tells us that c has to be grater than 4.
Third inequality:
b + c > a
c > a - b
c > 10 - 14
c > -4
This tells us that c has to be greater than -4. But we also know that c has to be greater than 4, so we take 4 as the minimal value for c.
Final answer:
4 < c < 24
Third side has to be greater than 4 and less than 24.
The number of bacteria in a culture is increasing according to the law of exponential growth. The initial population is 240 bacteria, and the population after 9 hours is double the population after 1 hour. How many bacteria will there be after 4 hours? (Round your answer to the nearest whole number.)
Answers
Answer:
339
Step-by-step explanation:
Exponential Function
General form of an exponential function: [tex]y=ab^x[/tex]
where:
a is the initial value (y-intercept)b is the base (growth/decay factor) in decimal formx is the independent variabley is the dependent variableIf b > 1 then it is an increasing function
If 0 < b < 1 then it is a decreasing function
Given information:
a = 240 (initial population of bacteria)x = time (in hours)y = population of bacteriaTherefore: [tex]y=240b^x[/tex]
To find an expression for the population after 1 hour, substitute x = 1 into the found equation:
[tex]\implies y=240b^1[/tex]
[tex]\implies y=240b[/tex]
We are told that the population after 9 hours is double the population after 1 hour. Therefore, make y equal to twice the found expression for the population after 1 hour, let x = 9, then solve for b:
[tex]\implies 2(240b)=240b^9[/tex]
[tex]\implies 480b=240b^9[/tex]
[tex]\implies 480=240b^8[/tex]
[tex]\implies 2=b^8[/tex]
[tex]\implies b=\sqrt[8]{2}[/tex]
[tex]\implies b=2^{\frac{1}{8}}[/tex]
Therefore, the final exponential equation modelling the given scenario is:
[tex]\implies y=240(2^{\frac{1}{8}})^x[/tex]
[tex]\implies y=240(2)^{\frac{1}{8}x}[/tex]
To find how many bacteria there will be after 4 hours, substitute x = 4 into the found equation:
[tex]\implies y=240(2)^{\frac{1}{8}(4)}[/tex]
[tex]\implies y=240(2)^{\frac{1}{2}}[/tex]
[tex]\implies y=339 \:\: \sf (nearest\:whole\:number)[/tex]
Therefore, there will be 339 bacteria (rounded to the nearest whole number) after 4 hours.
