Answer:
P(6,-π/6) = (6, -π/6 + 2nπ) and P(-6,-π/6) = (-6, -π/6 + (2n+1)π)
Step-by-step explanation:
We need to find all polar coordinates of
[tex]P = (6,\frac{-\pi }{6})[/tex]
The polar coordinates of any point can be reresented by (r,Ф)
where
(r,Ф) = (r, Ф+2nπ) where n is any integer and r is positive
and
(r,Ф) = (-r, Ф+(2n+1)π) where n is any integer and r is negative.
So, in the question given, r = 6 and Ф = -π/6
So, Polar coordinates will be:
P(6,-π/6) = (6, -π/6 + 2nπ) where where n is any integer and r is positive
and
P(-6,-π/6) = (-6, -π/6 + (2n+1)π) where n is any integer and r is negative.
Answer:
All polar coordinates of point P are [tex](6,-\frac{\pi}{5}+2n\pi)[/tex] and [tex](-6,-\frac{\pi}{5}+(2n+1)\pi)[/tex], where n is an integer.
Step-by-step explanation:
The given point is
[tex]P=(6,-\frac{\pi}{5})[/tex]
If a point is [tex]P=(r,\theta)[/tex], then all polar coordinates of point P are defined as
[tex](r,\theta)=(r,\theta+2n\pi)[/tex]
[tex](r,\theta)=(-r,\theta-(2n+1)\pi)[/tex]
where n is an integer.
In the given point [tex]r=6[/tex] and [tex]\theta=-\frac{\pi}{5}[/tex]. So all polar coordinates of point P are defined as
[tex](6,-\frac{\pi}{5})=(6,-\frac{\pi}{5}+2n\pi)[/tex]
[tex](6,-\frac{\pi}{5})=(-6,-\frac{\pi}{5}+(2n+1)\pi)[/tex]
Therefore all polar coordinates of point P are [tex](6,-\frac{\pi}{5}+2n\pi)[/tex] and [tex](-6,-\frac{\pi}{5}+(2n+1)\pi)[/tex], where n is an integer.
The diameter of the base of the cone measures 8 units.
The height measures 6 units.
What is the volume of the cone?
Answer: [tex]32\pi[/tex] cubic units.
Step-by-step explanation:
You can use this formula for calculate the volume of a cone:
[tex]V_{cone}=\frac{1}{3}\pi r^2h[/tex]
Where "r" is the radius and "h" is the height.
You know that the diameter of the base of the cone measures 8 units, then, the radius can be found by dividing the diameter by 2:
[tex]r=\frac{8units}{2}\\\\r=4units[/tex]
Since you already know that height and the radius, you can substitute them into the formula. Then, the volume of this cone is:
[tex]V_{cone}=\frac{1}{3}\pi (4units)^2(6units)[/tex]
[tex]V_{cone}=32\pi \ units^3[/tex]
Answer: 48
Step-by-step explanation:
Heo started to solve the quadratic equation (x + 2)2 – 9 = –5. He added 9 to both sides and the resulting equation was (x + 2)2 = 4. Next, he took the square root of each side. Which was the resulting equation of that step?
Step-by-step explanation:
After taking the square root:
x + 2 = ±2
Notice that x + 2 can also equal -2, because (-2)² = 4.
According to the diagram below, which similarity statements are true?
it can be concluded that [tex]\( \angle BAD = \angle CBD \) and \( \angle ABD = \angle BCD \).[/tex] So, all three angles are equal.
From the given information:
[tex]\( \angle BAD + \angle ABD = 90^\circ \)[/tex]
[tex]\( \angle CBD + \angle BCD = 90^\circ \)[/tex]
[tex]\( \angle ABD + \angle DBC = 90^\circ \)[/tex]
[tex]\( \angle BAD + \angle BCD = 90^\circ \)[/tex]
We can see that:
[tex]\( \angle BAD + \angle ABD = \angle CBD + \angle BCD \)[/tex]
[tex]\( \angle BAD + \angle BCD = \angle ABD + \angle DBC \)[/tex]
[tex]\( \angle BAD + \angle BCD = \angle BAD + \angle BCD \)[/tex]
From equations (1) and (2), we can conclude that:
[tex]\[ \angle BAD + \angle ABD = \angle CBD + \angle BCD = \angle ABD + \angle DBC \][/tex]
This implies that [tex]\( \angle ABD = \angle CBD \) and \( \angle BCD = \angle DBC \).[/tex]
Therefore, it can be concluded that [tex]\( \angle BAD = \angle CBD \) and \( \angle ABD = \angle BCD \).[/tex] So, all three angles are equal.
The complete question is:
According to the diagram below, which similarity statements are true? Check all that apply.
A. △ ABDsim △ BCD
B. △ ABCsim △ BDC
C. △ ABCsim △ ADB
D. △ ABDsim △ ADC
The similarity statements that are true are:
A. [tex]\(\triangle ABD \sim \triangle BCD\)[/tex]C. [tex]\(\triangle ABC \sim \triangle ADB\)[/tex]D. [tex]\(\triangle ABC \sim \triangle BDC\)[/tex][tex]\(\triangle ABD \sim \triangle BCD\):[/tex]
- Since D lies on AC and [tex]\( \angle ADB = \angle BDC = 90^\circ \),[/tex] both [tex]\(\triangle ABD\)[/tex]and \[tex](\triangle BCD\)[/tex]share [tex]\( \angle ADB = \angle BDC \).[/tex]
- They both share the common angle [tex]\( \angle B \).[/tex]
- Therefore, [tex]\(\triangle ABD \sim \triangle BCD\).[/tex]
[tex]\(\triangle ABC \sim \triangle ADB\):[/tex]
- Both triangles share the angle [tex]\( \angle A \).[/tex]
- They both have right angles at [tex]\( \angle ABC = \angle ADB = 90^\circ \).[/tex]
- Therefore, [tex]\(\triangle ABC \sim \triangle ADB\).[/tex]
[tex]\(\triangle ABC \sim \triangle BDC\):[/tex]
- Both triangles share the angle [tex]\( \angle C \).[/tex]
- They both have right angles at [tex]\( \angle ABC = \angle BDC = 90^\circ \).[/tex]
- Therefore, [tex]\(\triangle ABC \sim \triangle BDC\).[/tex]
A survey of patients at a hospital classified the patients by gender and blood type, as seen in the two-way table. Gender Blood Type Male Female TOTAL A 105 93 198 B 99 84 183 O 160 140 300 AB 15 18 33 TOTAL 379 335 714 What percent of all patients are blood type AB? Round your answer to the nearest tenth of a percent. Type a numerical answer in the space provided. Do not include % or spaces in your answer.
Answer:
4.6
Step-by-step explanation:
We are given the results of a survey of patients at a hospital of their blood type, categorized by their gender.
We are to find the percentage of patients with blood type AB among all patients.
Number of patients with blood type AB = 33
Total number of patients = 714
Percent of patients with blood type AB = [tex] \frac { 3 3 } { 7 1 4 } \times 100 [/tex] = 4.6
The hypotenuse of a 45°-45°-90° triangle measures 128 cm. What is the length of one leg of the triangle?
Answer:
90.51 (rounded to the nearest hundredth)
Step-by-step explanation:
We can use trigonometry to figure this out
sin = opposite side / hypothenuse
The 45° angle's opposite side is a leg of the triangle
sin 45 = leg / 128
sin 45 * 128 = (leg / 128 ) * 128
0.7071 * 128 = leg
90.51 = leg
Answer:
b
Step-by-step explanation:
Phillip has a box of crayons. 45 are yellow, 12 are green, 25 are blue, and 7 are red. If Phillip selects a crayon at random, which color crayon would he be MOST likely to select? Why?
Answer:
yellow b/c there are the most
Step-by-step explanation:
You flip a fair coin what is the probability that it show heads on the first flip and it shows how tails on the second flip
Answer:
25% = 1/4
Step-by-step explanation:
It is 25% because there are 4 outcomes that you can get which are
HT, TH, HH, TT
H = Head
T = Tail
and out of the four outcomes, there is one that will be Head first and Tails on the second flip. So 1 out of 4 = 25%
compete the square to determine minum or maxuim value of function define by -x2+10x+5
Answer:
maximum value y = 30
Step-by-step explanation:
Given
- x² + 10x + 5
To complete the square the coefficient of the x² term must be 1
factor out - 1
= - (x² - 10x) + 5
To complete the square
add/subtract ( half the coefficient of the x- term )² to x² - 10x
= - (x² + 2(- 5)x + 25 - 25) + 5
= - (x - 5)² + 25 + 5
= - (x - 5)² + 30 ← in vertex form
y = a(x - h)² + k
where (h, k) are the coordinates of the vertex and a is a multiplier
Hence vertex = (5, 30)
The max/ min occurs at the vertex
Since a < 0 then vertex is a maximum
Hence maximum value is y = 30
Graph the system of equations on graph paper to answer the question.
{y=14x+3
{y=2x+10
What is the solution for this system of equations?
Answer:
(x, y) = (7/12, 11 1/6)
Step-by-step explanation:
A graph will show you the solution is near (x, y) ≈ (0.6, 11.2). For an exact answer, an algebraic solution is indicated.
Subtracting the second equation from the first gives ...
0 = 12x -7
7/12 = x
Substituting for x in the second equation gives ...
y = 2(7/12) +10 = 7/6 +10
y = 11 1/6
If the area of square 1 is 25 units square, and the area of square 2 is 16 units square, what is the perimeter of square 3
Answer:
12
Step-by-step explanation:
If square 1's area is 25, that must mean its side is 5
If square 2's area is 16, that must mean its side is 4
Since it looks like its going in order, square 3's side is 3, and since there's 4 sides to a square, the perimeter is 12.
Answer:
12 units²
Step-by-step explanation:
4^2+b^2=5^2
b=25-16
b^2=9
b=√9
b=3
there are 4 sides in a square so 3•4=12
Which expression is equivalent to 7a^2b + 10a^2b^2 + 14a^2b^3?
ab(7a^2 + 10ab + 14b^2)
a^2b(7 + 10b + 14b^2)
7a^2(b + 3b^2 + 7b^3)
7a^2b^3(b^2 + 3b + 7)
Answer:
a^2b(7 + 10b + 14b^2)
Use wolframalpha for math questions, or photomath!
To solve this, factor out a^2b from the expression.
For this case we must indicate an expression equivalent to:
[tex]7a ^ 2b + 10a ^ 2b ^ 2 + 14a ^ 2b ^ 3[/tex]
We must draw the common term of the three terms, we have:
[tex]a ^ 2b (7 + 10b + 14b ^ 2)[/tex]
So:
[tex]7a ^ 2b + 10a ^ 2b ^ 2 + 14a ^ 2b ^ 3 = a ^ 2b (7 + 10b + 14b ^ 2)[/tex]
Answer:
[tex]a ^ 2b (7 + 10b + 14b ^ 2)[/tex]
Option B
Tammy mixes the letters s, c, h, o, o, and l thoroughly. without looking, allen draws one letter. expressed as a fraction, decimal, and percentage, what is the probability that allen will not select a consonant? (1 point)
There are 6 letters: (4 consonants, 2 vowels)
The probability that Allen will not select a consonant is: (2/6 which simplifies to 1/3) because there are only 2 letters out of the 6 that are not consonants
Fraction: 1/3
Decimal: 0.3333
Percentage: 33.33%
The probability that Allen does not select a consonant is 1/3 as a fraction, 0.3333 as a decimal, and 33.33% as a percentage.
You've asked what the probability is that Allen will not select a consonant when drawing one letter from the mix of letters 's, c, h, o, o, l'. First, let's determine how many vowels and consonants there are in the selection. We have two vowels ('o', 'o') and four consonants ('s', 'c', 'h', 'l'). This gives us a total of six letters.
To find the probability of not selecting a consonant, we look at the likelihood of selecting a vowel instead, since those are the only two types of letters available. There are 2 vowels out of a total of 6 letters. The probability as a fraction is thus 2/6, which simplifies to 1/3.
To express this probability as a decimal, divide the numerator by the denominator: 1 divided by 3 equals approximately 0.3333. Finally, to express this as a percentage, multiply the decimal by 100 to get 33.33%.
In summary:
Fraction: 1/3Decimal: 0.3333Percentage: 33.33%PLEASE HELP ASAP 45 PTS + BRAINLIEST TO RIGHT/BEST ANSWER
The answer is C. I have the work for it in the image bellow
Hope this helped!
~Just a girl in love with Shawn Mendes
Find the volume of this figure.
Answer:
320 cubic units
Step-by-step explanation:
The volume of a pyramid is given by the formula ...
V = (1/3)Bh
where B is the area of the base, and h is the height.
Your pyramid has a rectangular base with edge lengths 8 units and 10 units. Hence the area of the base is ...
8×10 = 80 . . . . square units
The height is 12 units, so the volume formula gives the volume as ...
V = (1/3)(80)(12) = 320 . . . . cubic units
А
Subtotal
Male
11
20
Female
13 T 22 TOT 10 us
24 27 20 26
Subtotal
a. Fill in the blank cells by computing subtotals. In the last call of the bottom
5pts
place the sum of all the interior cells. What is the total number of the
participated in the survey?
b. P(male)
Answer:
a)
The total number of students who participated in the survey is 97
b)
P(male) = 0.5361
c)
P(A) = 0.2474
d)
The events male and TV show B are not mutually exclusive since 5 males prefer TV show B.
e)
The events female and TV show C are mutually exclusive since no female participant prefers TV show C
Step-by-step explanation:
a)
The total number of students who participated in the survey is obtained as;
Number of male participants + number of female participants
52 + 45 = 97
b)
P(male)
This is the probability that a randomly selected individual would be a male;
P(male) = ( number of male participants) / ( total participants)
= 52/97 = 0.5361
c)
P(A)
This is the probability that a randomly selected participant would prefer TV show A;
P(A) = ( participants who prefer TV show A) / (total participants)
= 24/97 = 0.2474
d)
Two events are said to be mutually exclusive if they cannot happen at the same time. Another word that means mutually exclusive is disjoint. If two events A and B are disjoint, then the probability of them both occurring at the same time is 0.
The events male and TV show B are not mutually exclusive since 5 males prefer TV show B. The probability is thus not 0.
e)
Two events are said to be mutually exclusive if they cannot happen at the same time. Another word that means mutually exclusive is disjoint. If two events A and B are disjoint, then the probability of them both occurring at the same time is 0.
The events female and TV show C are mutually exclusive since no female participant prefers TV show C. The probability is thus 0.
.) The terms 4, t, 9 are the start of a sequence.
Part A: If the sequence is arithmetic, what is the value of t?
Part B: If the sequence is geometric, what is the value of t?
Answer:
see explanation
Step-by-step explanation:
A
If the sequence is arithmetic then the common difference d is
d = t - 4 = 9 -t, that is
t - 4 = 9 - t ( add t to both sides )
2t - 4 = 9 ( add 4 to both sides )
2t = 13 ( divide both sides by 2 )
t = [tex]\frac{13}{2}[/tex] = 6 [tex]\frac{1}{2}[/tex]
----------------------------------------------------------------------
B
If the sequence is geometric then the common ratio r is
r = [tex]\frac{t}{4}[/tex] = [tex]\frac{9}{t}[/tex] ( cross- multiply )
t² = 36 ( take the square root of both sides )
t = [tex]\sqrt{36}[/tex] = 6
A: If the sequence is arithmetic, what is the value of t is 6.5.
B: If the sequence is geometric, what is the value of t is 6.
If the sequence is arithmetic then the common difference d is,
What is the arithmetic sequence?An arithmetic sequence is a sequence of numbers where the differences between every two consecutive terms are the same.
d = t - 4
d= 9 -t,
that is,
t - 4 = 9 - t ( add t to both sides )
2t - 4 = 9 ( add 4 to both sides )
2t = 13 ( divide both sides by 2 )
t =[tex]\frac{13}{2}[/tex] = 6.5
If the sequence is geometric then the common ratio r is
r = [tex]\frac{t}{4} =\frac{9}{t}[/tex] = ( cross- multiply )
t² = 36 ( take the square root of both sides )
t = [tex]\sqrt{36}[/tex]= 6
Therefore we get,
A: If the sequence is arithmetic, what is the value of t is 6.5.
B: If the sequence is geometric, what is the value of t is 6
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In this triangle, what is the value of x?
Enter your answer, rounded to the nearest tenth, in the box.
x =
To find x you can use cosine since you know the adjacent side and the hypotenuse. Remember that for cosine it is adjacent over hypotenuse
cos(x) = [tex]\frac{28}{72}[/tex]
cos(x) = [tex]\frac{7}{18}[/tex]
To find the x we must take the inverse of cosine:
[tex]cos^{-1}[/tex]([tex]\frac{7}{18}[/tex] = x
x = 67.1146...
x ≈ 67.1 degrees
Hope this helped!
~Just a girl in love with Shawn Mendes
Answer:
67.1
Step-by-step explanation:
i took the test
Someone, please help. Will Mark brainlist if the answer is correct.
Screenshot it below.
[tex]\bf \textit{equation of a circle}\\\\ (x- h)^2+(y- k)^2= r^2 \qquad center~~(\stackrel{}{ h},\stackrel{}{ k})\qquad \qquad radius=\stackrel{}{ r} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ (x-9)^2+y^2=4\implies (x-\stackrel{h}{9})^2+(y-\stackrel{k}{0})^2=2^2~\hfill \stackrel{center}{(9,0)} \\\\\\ (x-\stackrel{h}{3})^2+(y-\stackrel{k}{2})^2=4\implies (x-\stackrel{h}{3})^2+(y-\stackrel{k}{2})^2=2^2~\hfill \stackrel{center}{(3,2)}[/tex]
Check the picture below.
well, its radius didn't change, anyhow, you know what to check out.
Answer:
The circle moves left and up
Step-by-step explanation:
The circle's center changes from (9, 0) to (3,2) and the radius stayed the same.
what is the length of the diagonal of a non-regulation tennis court with length 20 feet and width 15 feet?
Answer:
25 feet
Step-by-step explanation:
Basically that non-regulation tennis court is a rectangle. You want to know the length of the diagonal. If you draw it on paper, you'll see that this then become 2 triangles... of which you have 2 sides, and are seeking the hypotenuse. So....
H² = A² + B²
H² = 20² + 15² = 400 + 225 = 625
H = 25 feet.
Answer:
The diagonal of a non-regulation tennis court = 25 feet
Step-by-step explanation:
Pythagorean theorem
Hypotenuse² = Base² + Height²
The tennis court is like a rectangle.
We can consider the court as made of two right angled triangle
To find the length of diagonal of court
Here base = 15 feet and height = 20 feet
Diagonal or hypotenuse can be written as,
Diagonal ² = Base² + Height²
= 15² + 20²
= 225 + 400
= 625
Diagonal = √625 = 25 feet
Therefore the diagonal of a non-regulation tennis court = 25 feet
please help me figure out the radius
Answer:
Step-by-step explanation:
<EAF is a central angle. Its measure is 12o
Arc EF has angular measure of 12 degrees as well.
The formula for the arc length in cm is
Arc length = (given arc angle/360) * 2 * pi * r
r = 30 cm which is given
Given arc angle = 12 degrees.
Arc length = (12/360) * 2* pi * r
Arc length = 1/30 * 2*pi * 30
Arc length = 6.28
A student rolls a number cube Whose six faces are numbered 1 through 6 what is the sample space for this experiment.
Answer:
It depends on how you define an outcome.
Usually, the outcomes are considered to be the number showing on the top face of the die (number cube) when it comes to rest on a horizontal surface. If that is how you define outcomes, then the sample space is the set of numbers 1–6: {1, 2, 3, 4, 5, 6}.
Step-by-step explanation:
"This experiment" covers a lot of territory. The student could be checking to see if the die stays on the table or falls to the floor. The student could be experimenting to see if the die changes color or makes a certain kind of noise. The experiment could involve the number of times the die rolls over to a new face before it comes to rest. Until the experiment is properly defined, the sample space is unknown.
If the experiment is to see what number shows on the top face of the die at rest on a horizontal surface, then the sample space is presumably the set of numbers 1 through 6.
A parking lot has sixty-one parking spaces numbered from 1 to 61. There are no cars in the parking lot when Jillian pulls in and randomly parks. What is the probability that the number on the parking space where she parks is greater than or equal to 22?
A. 61/21
B. 40/61
C. 61/40
D. 21/61
Answer:
B. 40/61
Step-by-step explanation:
There are 40 spots out of the 61 in the lot with numbers greater than or equal to 22. The probability of choosing one of them at random is 40/61.
_____
About the answer choices
61/21 and 61/40 are both numbers that are greater than 1. A probability is always a number between 0 and 1 (inclusive), so these answers can be rejected immediately.
There are 21 spots with numbers less than 22, so 21/61 is the probability of choosing one of those. That is not what the question is asking for.
Jackie contributed a batch of baklava to the school bazaar. There are 12 pieces in a batch, and each piece sells for $1.50. If all the baklava is sold, what will Jackie's total contribution to the school be?
Answer:
Jackie's total contribution the school would be $18.00
Step-by-step explanation:
Jackie's contribution to the school bazaar = a batch of baklava
No. of pieces in a batch = 12 pieces
Price of one piece = $1.50
CASE: All baklava is sold
No. of pieces of Baklava sold = 12 pieces
Price of 12 pieces of Baklava = no. of pieces * Price of one piece
= 12 * $1.50
= $18.00
Which expression represents the area of the triangle drawn below (4x-3) (2x-1) (4x-2) (3x-3)?
Answer:
option C
Step-by-step explanation:
Area of the triangle=4x²+4x+1
Mr. McGregor has discovered that a large dog can destroy his entire garden in 2 hours and that a small boy can do the same job in 1 hour. How long would it take the large dog and the small boy working together to destroy Mr McGregor's garden?
The large dog and small boy working together will take 2/3 hour to destroy Mr McGregor's garden.
Explanation:To solve this problem, we can add the rates at which the large dog and the small boy work together:
Large dog's rate = 1 garden / 2 hours = 1/2 garden per hour
Small boy's rate = 1 garden / 1 hour = 1 garden per hour
Combined rate = 1/2 garden per hour + 1 garden per hour = 3/2 gardens per hour
Now, we can use the combined rate to find the time it would take for the large dog and the small boy to destroy Mr. McGregor's garden:
Time = 1 garden / Combined rate = 1 garden / (3/2 gardens per hour) = 2/3 hour
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Can someone help me out?
Answer:
see below
Step-by-step explanation:
If you haven't memorized it so you know it already, it takes about 10 seconds with your calculator to discover that ...
arccos((√3)/2) = β = 30°
This fact matches only one answer choice.
Can you help me with these 2 questions?
Answer:
What school you go to
Step-by-step explanation:
step by step go catch friend and step by step friends
How to write an exponential function from a table
Answer:
Step-by-step explanation:
I'll show you with an example. I'll make a table and then show you how to find the exponential equation, ok?
x y
0 3
1 12
2 48
3 192
The standard form of an exponential equation is
[tex]y=a(b)^x[/tex]
We will take 2 (x,y) coordinates from our table and fit them into the equations to solve for a and b. Using the point (0, 3):
[tex]3=a(b)^0[/tex]
b to the 0 power is equal to 1, so the equation then becomes a(1) = 3 so a = 3. We will use the other x, y coordinate along with that a value to now solve for b:
[tex]12=3(b)^1[/tex]
b to the first is b, so now that equation becomes 3b = 12 and b = 4. Filling that info back into the standard form:
[tex]y=3(4)^x[/tex]
There you go!
Which explanation best describes how to solve this problem? Marcia covered her kitchen floor with twice as much tile as her bathroom floor. She tiled 15 square feet of her bathroom floor. She also tiled a 9-square-foot area by her front door. How many square feet of her house did Marcia tile altogether?
Answer:
54 ft^2
Step-by-step explanation:
Bathroom area is 15 ft^2
The kitchen is twice the bathroom
2*15 = 30 ft^2
Kitchen: 30 ft^2
Front door: 9ft^2
The total is all the areas added together
Total = bathroom + kitchen + front door
= 15 ft^2 + 30 ft^2 + 9 ft^2
= 54 ft^2
The fish population of Lake Collins is decreasing at a rate of 5% per year. In 2004 there were about 1,350 fish. Write an exponential decay function that models this situation. Then find the population in 2010.
Answer:
Part 1) The exponential function is equal to [tex]y=1,350(0.95)^{x}[/tex]
Part 2) The population in 2010 was [tex]992\ fish[/tex]
Step-by-step explanation:
Part 1) Write an exponential decay function that models this situation
we know that
In this problem we have a exponential function of the form
[tex]y=a(b)^{x}[/tex]
where
y ----> the fish population of Lake Collins since 2004
x ----> the time in years
a is the initial value
b is the base
we have
[tex]a=1,350\ fish[/tex]
[tex]b=(100\%-5\%)=95\%=0.95[/tex]
substitute
[tex]y=1,350(0.95)^{x}[/tex] ----> exponential function that represent this scenario
Part 2) Find the population in 2010
we have
[tex]y=1,350(0.95)^{x}[/tex]
so
For [tex]x=(2010-2004)=6\ years[/tex]
substitute
[tex]y=1,350(0.95)^{6}=992\ fish[/tex]
Final answer:
The exponential decay function for the fish population in Lake Collins is P(t) = 1350 * e^(-0.05t). Using this, the estimated fish population in 2010 is about 1,000.
Explanation:
We need to write an exponential decay function to model the decreasing fish population in Lake Collins and then use it to find the population in 2010.
To create an exponential decay function, we can use the formula P(t) = P0 * e^(rt), where:
P(t) is the population at time t,
P0 is the initial population,
r is the rate of decay (as a negative value), and
t is the time in years since the initial count.
Given the initial population of 1,350 fish in 2004 and a decay rate of 5% per year, we can write the function as:
P(t) = 1350 * e^(-0.05t)
To find the population in 2010, we first calculate the time passed since 2004, which is 6 years. Thus, t = 6:
P(6) = 1350 * e^(-0.05*6)
Calculating this gives us:
P(6) ≈ 1350 * e^(-0.3) ≈ 1350 * 0.740818 ≈ 1000 (rounded to the nearest whole number)
Therefore, the fish population in 2010 was approximately 1,000 individuals.