If Matthew rolls a number cube 90 times, how many times can he expect it 2 poilto land on an odd number?
Answers
Solution
The total face of a cube is 6
Possible odd 1,3,5
[tex]Probability\text{ =}\frac{Possible\text{ outcome}}{Total\text{ outcome}}[/tex]The cube is numbered one through six, and the odd numbers are therefore 1,3,5. So the probability of rolling an odd number in any given single roll is
[tex]\frac{3}{6}=\frac{1}{2}[/tex]Multiplying this probability by the number of times she rolls the cube gives
[tex]\frac{1}{2}\times90=45[/tex]The final answer
[tex]45[/tex]Related Questions
If a single card is drawn from a standard 52-card deck, in how many ways could it be a diamond or a face card (a face card is a Jack, Queen, or King)?A. 4B. 21C. 22D. 13
Answers
The number of diamond cards is 13.
The number of face card is 12.
There are 3 face cards which are of diamond.
Thus, total number of non-face diamond cards is 13-3=10.
Thus, the requred number of ways are =12+10=22.
Thus, option (C) is correct.
solve system of equations by substution { y=3x+19{ y=5x+33
Answers
Given:
[tex]\begin{gathered} y=3x+19\ldots\ldots\ldots(1) \\ y=5x+33\ldots\ldots\ldots(2) \end{gathered}[/tex]To solve: The system of equations by substitution
Explanation:
Substituting equation (1) in equation (2), we get
[tex]\begin{gathered} 3x+19=5x+33 \\ 3x-5x=33-19 \\ -2x=14 \\ x=-7 \end{gathered}[/tex]Substitute the x value in equation (1), we get
[tex]\begin{gathered} y=3(-7)+19 \\ y=-21+19 \\ y=-2 \end{gathered}[/tex]Final answers: The solutions are,
[tex]\begin{gathered} x=-7 \\ y=-2 \end{gathered}[/tex]second blank has the option of , the same verticle asymptote as function h, vertical asymptote at x=-7, vertical asymptote at x=-5, and vertical asymptote at x=3
Answers
Given:
The graph is g(x) is given and the function h(x) is,
[tex]h(x)=g(x+5)[/tex]To classify the asymptotes:
Since the translated transformation of 5 units left,
There are no changes in the horizontal asymptote.
But, the vertical asymptote is,
[tex]\begin{gathered} x=-2-5 \\ =-7 \end{gathered}[/tex]Thus, the graph h(x)=g(x+5) has the same horizontal asymptote as the function g
Find from first principles the derivative of f(x)= root of X with respect to x
Answers
To find:
The derivative of function f(x) using the first principle.
[tex]f(x)=\sqrt{x}[/tex]Solution:
By the first principle, the derivative of the function f(x) is given by:
[tex]f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}[/tex]So, the derivative of the given function can be obtained as follows:
[tex]\begin{gathered} f^{\prime}(x)=\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h} \\ =\lim_{h\to0}\frac{\sqrt{x+h}-\sqrt{x}}{h}\times\frac{\sqrt{x+h}+\sqrt{x}}{\sqrt{x+h}+\sqrt{x}} \\ =\lim_{h\to0}\frac{x+h-x}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{h}{h(\sqrt{x+h}+\sqrt{x})} \\ =\lim_{h\to0}\frac{1}{(\sqrt{x+h}+\sqrt{x})} \\ =\frac{1}{\sqrt{x+0}+\sqrt{x}} \\ =\frac{1}{2\sqrt{x}} \end{gathered}[/tex]Thus, the derivative of the given function is:
[tex]f^{\prime}(x)=\frac{1}{2\sqrt{x}}[/tex]if andrew can run 60 meters in 6 seconds, how many meters can he run in 1 second?
Answers
Explanation
Step 1
the sp
Santa is saving money for a new bike that costs $175.62. He already has $39.59. How much more does he need to save before he can buy a bike?
Answers
Given
Santa is saving money for a new bike that costs $175.62. He already has $39.59.
Answer
Total money needed = 175.62 - 39.59 = $136.03
How many distinct rearrangements of the letters in 'PUYGPGPYYUG' are there?
Answers
The answer is 92400
To solve this, we can count how many letters we have. There are 11 letters.
If those 11 letters were different from each other, the answer would be 11!
But we have letters that repeats:
3 P's
3 Y'2
3 G'2
2 U's
Since we want to know the quantity of distinct arrangements, we can divide by the repetition. This means:
[tex]\begin{gathered} 11!\text{ = total combinations} \\ \frac{11!}{3!\cdot3!\cdot3!\cdot2!}=\text{total distinct combinations} \end{gathered}[/tex]We divide by 3! times 3! times 3! times 2!, because we have 3P's, 3Y's, 3G's and 2U's
Then on the calculator write the division and give us the answer 92400
Find the domain and range. Select the correct symbols to indicate interval notation.If a number is not an integer then round to the nearest hundredth.
Answers
Remember that
The Domain is the set of all the input values, which are the x-coordinate of each ordered pair (the first number in each pair).
The Range is the set of all output values, which are the y-coordinate of each ordered pair (the second number in each pair).
so
In this problem
The domain is the interval [-5,3)
The range is the interval [0,2]
10. f(x) = 2x+5 if x < 4 1x² + 3x if x 24 LX D = R =
Answers
The domain of the given piecewise function are all the values that x can take.
These are:
x < -4 and
[tex]x\ge-4[/tex]It can be written as:
[tex](-\infty,\infty)[/tex]The range is:
[tex](-\infty,-3)\cup\lbrack-\frac{9}{4},\infty)[/tex]The perimeter of a rectangle is 32 meters and the length is 4 meters longer than width
Answers
Given:
The perimeter of a rectangle is 32 meters and the length is 4 meters longer than the width
Let, x = the length of the rectangle
And, y = the width of the rectangle
So, we have the following system of equations:
[tex]\begin{gathered} 2x+2y=32\rightarrow(1) \\ x-y=4\rightarrow(2) \end{gathered}[/tex]We will use the method of substitution to solve the system
So, from equation 2:
[tex]x=y+4\rightarrow(3)[/tex]substitute with (x) from equation (3) intp eqaution (1)
[tex]2(y+4)+2y=32[/tex]solve the equation to find (y):
[tex]\begin{gathered} 2y+8+2y=32 \\ 4y+8=32 \\ 4y=32-8 \\ 4y=24 \\ y=\frac{24}{4}=6 \end{gathered}[/tex]Now, substitute with (y) into equation (3) to find (x):
[tex]x=y+4=6+4=10[/tex]So, the answer will be:
The length of the rectangle = 10 m
The width of the rectangle = 6 m
Woad holder mange en windows-17Grado2rean, thetwo w Wawie kan 137( wow month am wa mama w War[mmer
Answers
a) The temperature at Milwaukee by 6am is given to be -5 degree fahrenheit.
By noon, this temperature has risen to 13 degree fahrenheit, thus this implies;
[tex]-5+13=8^0F[/tex]Hence, the temperature at Milwaukee by noon will be 8 degree fahrenheit
b) Temperature in Winnipeg = -17 degree fahrenheit
Temperature in Orlando = 61 degree fahrenheit
Temperature in Winnipeg is lower than in Orlando implies;
[tex]61-(-17)=61+17=78^0F[/tex]Hence, the temperature in Winnipeg is 78 degree fahrenheit lower than in Orlando
у20f15(4,12)10g(4,6)(-2,0)(0,25) 2,4,(3,5)(2,0)0,47Use the graph off and g.Find (gof)(2).ما
Answers
The first step is to find the fuctions, f(x) and g(x)
Since function f(x) is represented by the curve, it's a quadratic function. The curve cuts the x axis at x = 2 and x = - 2. Thus, the factors are (x + 2) and (x - 2). The quadratic function would be
(x + 2)(x - 2)
= x^2 - 2x + 2x - 4
f(x) = x^2 - 4
Since the function g(x) is represented by a straight line, it is a linear function. We would represent the function in the slope intercept form which is expressed as
y = mx + c
where
m = slope
c = y intercept
To find slope, the formula is
m = (y2 - y1)/(x2 - x1)
From the given points,
when x1 = - 2, y1 = 0
when x2 = 2, y2 = 4
m = (4 - 0)/(2 - - 2) = 4/(2 + 2) = 4/4
m = 1
the y intercept is the value of y when x = 0. Thus, c = 2
The function is
g(x) = x + 2
To find (g o f)(x), we would substitute function f into function g. Thus,
gof(x) = x^2 - 4 + 2
gof(x) = x^2 - 2
To find gof(2), we would substitute x = 2 into gof(x). It becomes
gof(2) = 2^2 - 2 = 4 - 2
gof(2) = 2
the answer and how to figure questions out like this!
Answers
In order to find the exponential regression we are going to select some values of the given data.
STEP 1An special value is when x=0.
On the table we can see that when x=0 then y=9
Replacing x by 0 in the given choices, we have that:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^0=8.04\cdot1 \\ =8.04 \end{gathered}[/tex][tex]\begin{gathered} B\text{.} \\ y=3.02\cdot3.67^x \\ \downarrow \\ y=3.02\cdot3.67^0=3.02\cdot1 \\ =3.02 \end{gathered}[/tex][tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^0=6.61\cdot1 \\ =6.61 \end{gathered}[/tex][tex]\begin{gathered} D\text{.} \\ y=2.27\cdot2.09^x \\ \downarrow \\ y=2.27\cdot2.09^0=2.27\cdot1 \\ =2.27 \end{gathered}[/tex]Observing the results we have that the two choices with closer results to 9 are A (with 8.04) and C (with 6.61)
STEP 2Now, we are going to select two additional values from the table in order to find which is the best answer: A or C.
Let's take x=1.
When x = 1, then y=10.
Replacing on the equation A we have:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^1=8.04\cdot0.98 \\ =7.879 \end{gathered}[/tex]and for the equation C:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^1=6.61\cdot1.55 \\ =10.2455 \end{gathered}[/tex]For x=1, the nearest result is from the equation C.
Let's verify what happens when x=2.
When x=2 then y=16. Replacing on the equation A we have:
[tex]\begin{gathered} A\text{.} \\ y=8.04\cdot0.98^x \\ \downarrow \\ y=8.04\cdot0.98^2 \\ =7.7216 \end{gathered}[/tex]and for the equation C:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^2 \\ =15.88 \end{gathered}[/tex]Again, for x=2, the nearest result is from the equation C.
Then, we can conclude that the best candidate is equation C.
We could try other values of x to double check:
[tex]\begin{gathered} C\text{.} \\ y=6.61\cdot1.55^x \\ \downarrow \\ y=6.61\cdot1.55^2 \\ =15.88 \end{gathered}[/tex]Answer the statistical measures and create a box and whiskers plot for the followingset of data.4,4,5,5,5,6,6, 8, 9, 10, 12, 14, 14, 15, 17
Answers
Notice that the set of data is already ordered from lowest to highest, therefore the minimum value is 4 and the maximum value is 17.
The median is 8, the first quartile Q₁=5, and Q₃=14.
Then, the corresponding box and whiskers plot is:
Select the following sentence that represents the equation below:
3(n2+1)=3n+12
Responses
The sum of the product of two and a number plus one times three is equal to twelve more than the product of three and the same number.
The sum of the product of two and a number plus one times three is equal to twelve more than the product of three and the same number., EndFragment,
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number., EndFragment,
The quotient of a number and two increased by one is equal to twelve more than the quotient of three and the same number.
The quotient of a number and two increased by one is equal to twelve more than the quotient of three and the same number., EndFragment,
Three times the sum of a number divided by two and one is equal to three times the same number increased by twelv
Answers
The answer is, Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
Three times the sum of twice a number and one is equal to twelve less than the product of three and the same number.
What is a quotient?It has a wide spread throughout the Mathematics. It is referred to as the integer part of a division, or as the fraction, or as a ratio. It is used when indicating the presence or the degree of a characteristic in something or by someone. Quotient is the result of a division. It is obtained when we divide one number by another. Quotient means how many times. And it is derived from the Latin.
To know more about quotient, visit:
https://brainly.com/question/16134410
#SPJ1
The arcade charges $125.00 to reserve the location, and then $15.00 per person. Which expressionrepresents the total cost for any number of people n?
Answers
Let 'n' represent the number of people. Sin the arcade charces 15 per person, then we have the following first term:
[tex]15n[/tex]since the charge to reserve is $125, we have:
[tex]15n+125[/tex]thus, if 'c' represents the total cost, then the expression to represent this situation is:
[tex]c=15n+125[/tex]Use elimination to solve each system of equations.4x + y = 233x - y = 12Use elimination to solve each system of equations.4x + y = 233x - y = 12
Answers
ANSWER
The solution is (5, 3)
EXPLANATION
To use elimination method, we have to subtract (or add) one equation from the other, in order to eliminate one of the variables. Then we'll have one equation with one variable. We solve it for that variable and then replace into one of the equations of the system to solve for the eliminated variable.
In these equations, we have +y in the first one and -y in the second one. We can add both equations to eliminate y:
[tex]\begin{gathered} (4x+y)+(3x-y)=23+12 \\ 4x+3x+y-y=35 \\ 7x=35 \end{gathered}[/tex]Solving for x:
[tex]\begin{gathered} 7x=35 \\ x=\frac{35}{7} \\ x=5 \end{gathered}[/tex]Now, we have to replace x = 5 into one of the equations and solve for y. Replacing in the first equation:
[tex]\begin{gathered} 4x+y=23 \\ 4\cdot5+y=23 \\ 20+y=23 \\ y=23-20 \\ y=3 \end{gathered}[/tex]The solution to this system is (5, 3)
Martha opened a savings account and deposited 400.00 the account earns 1%interest compounded annually what is the balance after 3 years
Answers
p = 400.00
r = 1% = 1/100 = 0.01
t = 3 years
Therefore,
[tex]\begin{gathered} Amount=p(1+\frac{r}{n})^{nt} \\ \text{Amount}=400(1+\frac{0.01}{1})^{1\times3} \\ \text{Amount}=400\times1.030301 \\ \text{Amount}=\text{ \$}412.1204 \\ \text{Amount}=\text{ \$412.12} \end{gathered}[/tex]After three years the amount will be = $412.12
Inderstanding ocabulary Are 23 and 24 adjacent angles? Explain. 1. 2. 3. 4 3 4 3 4 3 . Reasoning Does every angle have a complement? Explain. ises For more exercises, see Extra Skills and Word Problem me a pair of vertical and adjacent angles in each figure. Find m21.
Answers
By definition, two angles are adjacent if they share one side and the vertex.
To determine if ∠3 and ∠4 are adjacent, you have to look at each image and determine if they share the vertex and one side:
1.
In this image ∠3 and ∠4 share, the vertex but they do not share one side, this indicates that these angles are not adjacent.
2.
In this image ∠3 and ∠4 share the vertex and one side (blue line), which indicates that they are adjacent angles.
3.
In this image ∠3 and ∠4 share one side (blue line) but each angle has its own vertex (green dots). You cannot conclude these angles are adjacent.
4.
Two angles are complementary of they add up to 90º, they don't necesarly have to be adjacent.
Any acute angle, meaning, any angle that measures less than 90º, has a complement.
For example, you have the following angles:
- If both ∠1 and ∠2 are acute angles and complementary, then we know that they add up to 90º:
[tex]\angle1+\angle2=90º[/tex]-For example, ∠1= 46º, then you can determine the measure of ∠2 as follows:
[tex]\begin{gathered} \angle2=90º-\angle1 \\ \angle2=90-46 \\ \angle2=44º \end{gathered}[/tex]Both ∠1=46º and ∠2=44º are acute and add up to 90º
-If one of the angles is a right angle, for example, ∠2=90º, then no matter what measure does ∠1 take, they will never add up to 90º.
We can say that right angles do not have a complement.
-If one of the measures of the angles is more than 90º (it is an obtuse angle), let's say, for example, ∠1= 124º, no matter what measure ∠2 has, if you add both angles, they will never add up to 90º.
So we can say that obtuse angles have no complements.
In conclusion, not all angles can have a complement, only acute angles have complements.
A teacher asks his students to use the Addition Property of Equalityto write an equation equivalent to x - 9 = 11 Antonio writes* - 9 + 9 + 11 + 9. Stefan writes x - 9+2 -11 + 2 Have bothstudents followed the teacher's instructions? Explain your reasoning
Answers
Both students followed the teacher's instruction. The addition property of equality states that if 2 numbers x and y are equal x=y, then, x+a=y+a.
In this case both students applied the property correctly. One of them added 9 to both sides of the equation and the other one added 2 to both sides.
The answer is yes, both followed the teacher's instruction.
Please I need help on this ASAP
Answers
The value of the function h(x) in part (a) is equal to h(x) = 2x.
The value of the function h(x) in part (b) is equal to h(x) = 5x/2 - 2.
How to determine the function h(x)?In this exercise, you are required to calculate the value of the function h(x), which is a product of the addition of function f(x) and function g(x). This ultimately implies that, the value of function h(x) can be calculated by adding function f(x) and function g(x) together.
For the first part (a), the value of function h(x) would be calculated as follows:
Function h(x) = Function f(x) + Function g(x)
Substituting the given parameters into the formula, we have;
Function h(x) = (x + 4) + (x - 4)
Function h(x) = x + 4 + x - 4
Function h(x) = 2x
For the second part (b), the value of function h(x) would be calculated as follows:
Function h(x) = Function f(x) + Function g(x)
Substituting the given parameters into the formula, we have;
Function h(x) = (2x - 4) + (x/2 + 2)
Function h(x) = 2x - 4 + x/2 + 2
Function h(x) = 5x/2 - 2.
Read more on function here: brainly.com/question/3632175
#SPJ1
The shorter leg of a 30°-60°-90° triangle measures 9sqrt3 inches. What is the length of the longer leg? OA. 27 inches OB. 27sqrt3 inches OC. 18 inches OD. 18sqrt3 inches
Answers
We know that the proportion of the sides of a 30°-60°-90° triangle is:
The shorter leg is K, then:
[tex]K=9\sqrt[]{3}\text{ in}[/tex]Using this result, we can calculate the length of the longer leg:
[tex]\begin{gathered} \sqrt[]{3}K=\sqrt[]{3}\cdot9\cdot\sqrt[]{3}=9\cdot3 \\ \Rightarrow=27\text{ in} \end{gathered}[/tex]Look at the four company logos below.VolkswagenLincolnLexusRed Cross0♡+The logo for Volkswagen haslines of symmetry.The logo for Lincoln haslines of symmetry.The logo for Lexus haslines of symmetry.The logo for Red Cross has4lines of symmetry.:: 0.: 1:: 2
Answers
In this case, we'll have to carry out several steps to find the solution.
Step 01:
Data:
Figures logos
Lines of symmetry = ?
Step 02:
We must analyze the logos to find the solution.
Volkswagen ===> 1 line of symmetry
Lincoln ===> 2 lines of symmetry
Lexus ===> 0 lines of symmetry
Red Cross ===> 4 lines of symmetry
That is the solution.
Answer:
VW: 1
Lincoln: 2
Lexus: 0
Red Cross: 2
Step-by-step explanation:
VW can be split in half once, and have the same thing son both sides.
Lincoln be split in half horizontally and vertically and be identical on both sides
Lexus cannot be split in half at all and be identical.
The red cross can be split horizontally and vertically and still have identical pieces.
-Hope this helped
How can we find the solutions for an equation like 6 cos x = 2 in the interval 0 to 2π?
Answers
Answer:
[tex]x=\arccos(1/3), 2\pi-\arccos(1/3)[/tex]
Step-by-step explanation:
[tex]6\cos x=2 \\ \\ \cos x=1/3 \\ \\ x=\arccos(1/3), 2\pi-\arccos(1/3)[/tex]
HELP ASAP!!! THE BEST ANSWER GETS BRAINLIST! (15 POINTS)
(07.05A HC)
The following table shows the values of y for different values of x:
x y
0 -5
1 0
2 5
Which statement best explains whether the table represents a linear function or a nonlinear function?
It represents a linear function because its points are on a straight line.
It represents a linear function because its points are not on a straight line.
It represents a nonlinear function because its points are on a straight line.
It represents a nonlinear function because its points are not on a straight line.
Answers
The first choice is the correct answer.
It represents a linear function because its points are on a straight line
Solve the inequalityNine times c is less than -15.
Answers
Nine times c is less than -15.
Can be written as
[tex]\begin{gathered} 9c<-15 \\ \Rightarrow \\ \frac{9c}{9}<\frac{-15}{9} \\ \Rightarrow c<\frac{-5}{3} \\ \Rightarrow c<-1\frac{2}{3} \end{gathered}[/tex]the area of the shaded circular sector is equal to 30. The radius of the circle is 10. Find the measure of the central angle (in degrees)
Answers
Given:
There are given that the area of the shaded circular sector is:
[tex]30\pi[/tex]Explanation:
To find the central angle, we need to use the formula of area of the sector:;
So,
From the formula of area of the sector:
[tex]Area\text{ of sector=}\frac{central\text{ angle}}{360^{\circ}}\times\pi r^2[/tex]Then,
Put the value of area and radius into the above formula;
So,
[tex]\begin{gathered} Area\text{ of sector=}\frac{central\text{ angle}}{360^{^{\circ}}}\times\pi r^2 \\ 30\pi=\frac{centralangle}{360}\pi\times(10)^2 \end{gathered}[/tex]Then,
[tex]\begin{gathered} 30\pi=\frac{centralangle}{360}\pi(10)^{2} \\ 3=\frac{centralangle}{36} \\ central\text{ angle=36}\times3 \\ central\text{ angle=108}^{\circ} \end{gathered}[/tex]Final answer:
hence, the central angle is 108 degrees.
A bag of Mand Ms contains 5 yellow, 11 red, 4 green, 12 blue and 7 brown candies.What is the probability that a red or brown candy is pulled from the bag?A. 18/39B. 18/78C. 9/20D. 77/1521
Answers
In this question, we need to find the probability of pulling a red or brown candy from a bag.
We know that the bag contains the next amount of candies:
• 5 yellow candies
,• 11 red candies
,• 4 green candies
,• 12 blue candies
,• 7 brown candies
Therefore, in total, we have 39 candies.
Then the probability of pulling a red candy is:
[tex]P(\text{red)}=\frac{11}{39}[/tex]The probability of pulling a brown candy is:
[tex]P(\text{brown)}=\frac{7}{39}[/tex]Now, we know that the general formula for the probability of two events is given by:
[tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]However, in this case, we do not have any probability that both events happen at the same time - in other words, they are mutually exclusive events. Therefore, we have:
[tex]\begin{gathered} P(R\cup B)=P(R)+P(B)-P(R\cap B) \\ P(R\cup B)=\frac{11}{39}+\frac{7}{39}=\frac{18}{39} \\ P(R\cup B)=\frac{18}{39} \end{gathered}[/tex]Therefore, in summary, the probability that a red or brown candy is pulled from the bag is 18/39 (option A.)
write a quadratic function whose graph has the given characteristicsvertex: (1,2)Point: (3,6)
Answers
Solution
For this case the general expression for a parabola is given by:
y- k= a(x-h)^2
From the info given we know that h = 1 and k= 2
And replacing we got:
y -2 = a(x-1)^2
And replacing the point given we got:
6-2 = a(3-1)^2
4= 4a
a=1
And the equation would be given by:
y-2 = (x-1)^2
16. The table below shows the population of California from 2010 to 2019.
Answers
The Solution:
The Regression Model that best fits the data given in the question is
[tex]P(t)=\frac{a}{1+e^{-bt}}[/tex]From the Desmos plotter analysis attached above, we have that
[tex]\begin{gathered} a=74.907\approx74.91 \\ b=0.01397\approx0.01 \end{gathered}[/tex]So, by substituting the values of the parameters, we get the required logistic Regression Model is given below:
[tex]P(t)=\frac{74.91}{1+e^{-0.01t}}[/tex]b. The model predicts that the population of California in 2025 will be:
[tex]\begin{gathered} \text{From 2010 to 2025 is 25 years.} \\ So,\text{ t=25 years.} \\ S\text{ubstituting 25 for t in the regression model, we get} \end{gathered}[/tex][tex]P(t)=\frac{74.91}{1+e^{-0.01(25)}}=\frac{74.91}{1+0.77088}=\frac{74.91}{1.77088}=42.1127\approx42.1\text{ million people.}[/tex]So, the population of California in 2025 will be 42.1 million people.
c. To find when the model predicts that the population of California will be 40 million,
we shall substitute 40 (in millions) for P in the model, and find t as below:
[tex]40=\frac{74.91}{1+e^{-0.01t}}[/tex]Cross multiplying, we get
[tex]\begin{gathered} 40(1+e^{-0.01t})=74.91 \\ \text{Dividing both sides by 40},\text{ we have} \\ 1+e^{-0.01t}=\frac{74.91}{40} \\ \\ 1+e^{-0.01t}=1.87275 \\ e^{-0.01t}=1.87275-1 \\ e^{-0.01t}=0.87275 \end{gathered}[/tex]Taking the ln of both sides, we get
[tex]\begin{gathered} \ln e^{-0.01t}=\ln 0.87275 \\ -0.01t=\ln 0.87275 \\ -0.01t=-0.136106 \\ \text{ Dividing both sides by -0.01, we get} \\ t=\frac{-0.136106}{-0.01}=13.61\approx14\text{ years} \end{gathered}[/tex]So, 14 years from 2010 will be in the year 2024
d. According to the model, the carrying capacity for California's capacity is 74.9 million people.