Answer:
Step-by-step explanation:
rnw
.a local blood center needs donors and has advertised on the college campus. faculty, staff and students all donated during the drive. many also volunteered to help set up , hand out materials and clean up at the end of the event. there were a total of 500 people surveyed about their involvement that day. two hunded seventeen of them gave blood. one hundred fifty-six helped with setting up and cleaning up, as well as handing out materials. if ninety-three people both helped and donated, find out how many people neiher donated nor helped.
Answer:
There are 500 total people. 215 gave blood and 159 helped set up, giving a total of 374. However, since 89 people did both, we counted those 89 people twice. So we subtract 89 to get 285 total volunteers.
500-285=215 did not volunteer
There were 220 people that neither donated nor helped.
To find out how many people neither donated nor helped, we can use the formula for the union of two sets:
A∪B = A + B - A∩B.
In this case, A is the number of people who donated, B is the number of people who helped, and A∩B is the number of people who both donated and helped.
Plugging in the given values, we get:
A∪B = 217 + 156 - 93 = 280
This tells us that there were 280 people who either donated or helped. To find out how many people neither donated nor helped, we can subtract this number from the total number of people surveyed:
500 - 280 = 220
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If someone could help I would really appreciate it. The scale on a map is 1: 200,000. The length of a road on the map is 4 cm What is the length of the road in real life? Give your answer in kilometres.
Answer: If the ratio is 1:200,000 then the length of the road in centimeters is 800,000 centimeters meaning the conversion of this to kilometers would be 8 kilometers
Step-by-step explanation: every 100k (100,000) cm in kilometers would be 1 kilometer so take that and for the 200,000 times 4 is 800,000 which means the answer is 8 kilometers
________________________________________________________
I need help with this Quadratic equation-standard form
POINT A (5,5) POINT B (5,-5) Solve using two point form, express in standard form
Using two-point form, the equation of the line passing through the points A and B is x = 5.
The two-point form of a line is given by:
y - y₁ = [(y₂ - y₁)/(x₂ - x₁)] (x - x₁)
where (x₁, y₁) and (x₂, y₂) are two points on the line.
Given points A(5, 5) and B(5, -5), we can plug in the values into the equation:
y - 5 = [(-5 - 5)/(5 - 5)] (x - 5)
Simplifying the equation gives us:
y - 5 = (-10/0) (x - 5)
Notice that the slope (y₂ - y₁)/(x₂ - x₁) is equal to -10/0(indeterminate). This means that the line is a vertical line passing through x = 5.
The standard form of a line is Ax + By = C. So, the standard form of the equation of the line passing through points A and B is: x = 5.
Therefore, the equation of the line passing through the two points is x = 5.
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Which is greater, the number of bacteria, or the number of all the other animals in the table put together?
The number of bacteria or the number of all the other animals in the table put together is 7.8 x 10⁶.
To compare the number of bacteria to the number of all other animals in the world, we need to use some mathematical estimates. It's important to note that it's almost impossible to count the exact number of bacteria and animals in the world since they are found in vast numbers and diverse environments.
According to some scientific estimates, the number of bacteria on Earth is around 5 x 10³⁰, which is a massive number. In comparison, the total number of all other animals, including insects, birds, and mammals, is estimated to be around 7.8 x 10⁶.
In conclusion, the number of bacteria is much greater than the number of all other animals in the world. Although bacteria are tiny microorganisms, they are present in vast numbers and can be found in almost every environment.
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Starting time: 9:55 A. M. Elapsed time: 27 minutes
In the following question, Starting time: 9:55 A. M. Elapsed time: 27 minutes provide a snapshot of a specific moment in time and the amount of time that has passed since a particular event or task began.
You have provided two pieces of information: the starting time and the elapsed time.
The starting time is 9:55 A.M., which means that an event or task began at that time. The time is given in the 12-hour clock format, where A.M. stands for "ante meridiem" and refers to the period between midnight and noon.
The elapsed time is 27 minutes, which means that 27 minutes have passed since the event or task started. "Elapsed time" refers to the amount of time that has passed since the start of an event or task.
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A pack of sweets that are 81 inside, must be divided in a ratio of 2:1, how many will each get
The each of two individuals will get 54 and 27 sweets according to provided ratio.
Let us assume the amount of sweets each get be 2x and x. Now we know the sum is 81, so representing the information in equation form.
2x + x = 81
Performing addition on Left Hand Side of the equation
3x = 81
Rewriting the equation
x = 81/3
Performing division on Right Hand Side of the equation
x = 27
So, first person will get amount of sweets = 2×27
First person = 54
Amount of sweets for second person = 27
Thus, each will get 54 and 27 sweets.
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An accountant is modeling the annual tax expenditures, e, in thousands of dollars t years after january 1st, 2000 for a small business using two different models. Both of the accountant's models have tax expenditures of $5000 on january 1st, 2000. What is the value of e?
Both models assume that the tax expenditures are $5000 on January 1st, 2000.
The accountant is using two different models to estimate the tax expenditures for the small business. Both models assume that the tax expenditures are $5000 on January 1st, 2000. The accountant wants to estimate the tax expenditures, e, in thousands of dollars t years after January 1st, 2000.
Let's call the two models Model 1 and Model 2. Model 1 assumes that the tax expenditures increase by a fixed amount every year. Let's call this fixed amount a. Therefore, the tax expenditures, e, under Model 1 can be represented by the equation:
e = 5000 + at
Model 2 assumes that the tax expenditures increase at a constant percentage rate every year. Let's call this percentage rate r. Therefore, the tax expenditures, e, under Model 2 can be represented by the equation:
e = 5000(1 + r)ˣ
Now, let's substitute t = 0 into both equations to find the value of e on January 1st, 2000.
For Model 1:
e = 5000 + a(0) = 5000
For Model 2:
e = 5000(1 + r)⁰ = 5000
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How long would Debbie have to leave her money in two accounts, $5000 at 8.75% and $7000 at
9.5%, in order to earn $8820 in interest?
Assuming this is a simple interest situation:
The 8.75% account will earn (0.0875)(5000) each year.
$437.5 earned per year
The 9.5 % account will earn (0.095)(7000) each year
$665 earned per year
Together, these accounts earn $1102.5 per year.
If x is the number of years until the desired interest is earned, then you need to solve 1102.5x = 8820. Dividing by 1102.5 on both sides, you find x=8.
Again, assuming this is simple interest and not compound interest, it will take 8 years.
Consider a gamble X such that X = | -$100 with probability 0.4 (a) Is this a fair gamble? Verify your answer (b) Suppose that Anh has an initial wealth of $100 and her utility function over money is u(w) = Vw. Will she join this gamble? Justify your answer. (c) Suppose that Chris has an initial wealth of $0 and his utility function over money is u (w) = e ico. Will he join this gamble? Justify your answer. (d) Suppose that Beth has an initial wealth of $10000 and her utility function over money is u(w) = vw. Will she join this gamble? Justify your answer and explain difference from part (b)
The difference from part (b) is that Beth has a much higher initial wealth, so the negative expected value of the gamble has a smaller impact on her overall utility.
This is not a fair gamble because the expected value of the gamble is negative. The expected value of a gamble is calculated by multiplying the probability of each outcome by the value of that outcome and summing the results. In this case, the expected value is 0.4(-$100) = -$40. Since the expected value is negative, the gamble is not fair.
Anh will not join this gamble because her expected utility from the gamble is lower than her utility from her initial wealth. Her expected utility from the gamble is
0.4u(-$100) = 0.4v(-$100) = -40v.
Her utility from her initial wealth is
u($100) = v($100) = 100v.
Since -40v < 100v, her expected utility from the gamble is lower than her utility from her initial wealth, so she will not join the gamble.
Chris will not join this gamble because his expected utility from the gamble is lower than his utility from his initial wealth. His expected utility from the gamble is
0.4u(-$100) = 0.4e^(-100c) = -40e^(-100c).
His utility from his initial wealth is
u($0) = e^(0c) = 1.
Since -40e^(-100c) < 1, his expected utility from the gamble is lower than his utility from his initial wealth, so he will not join the gamble.
Beth will not join this gamble because her expected utility from the gamble is lower than her utility from her initial wealth. Her expected utility from the gamble is
0.4u(-$100) = 0.4v(-$100) = -40v.
Her utility from her initial wealth is
u($10000) = v($10000) = 10000v.
Since -40v < 10000v, her expected utility from the gamble is lower than her utility from her initial wealth, so she will not join the gamble.
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A dealer sold a car to a man and made a profit of 15percent .The man then sold it to a woman for 120175 naira at a loss of 50percent . how much did the dealer buy the car
The dealer bought the car, which he sold to a man and made a profit of 15 percent at $209,000.
How is the dealer's purchase price determined?To compute the dealer's purchase price, we use backward calculations and percentages.
First, the man's purchase price is determined as $240,350 since he incurred a loss of 50% amounting to $120,175.
By equating the proportional loss to the selling price, we can determine the purchase price paid by the man as $240,350.
Since the dealer generated a profit of 15% on $240,350, we determine his cost as $209,000 based on the profit percentage.
The dealer's profit percentage = 15%
The man's loss percentage = 50%
The selling price by the man to the woman = $120,175
The purchase price by the man = $240,350 ($120,175/50%)
The dealer's purchase price = $209,000 ($240,350/1.15)
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PLEASE HELP I WILL GIVE 30 points
Answer:
Your answer is Point T
Step-by-step explanation:
Answer:
s
Step-by-step explanation:
the missing side of each triangle. Round your answers to the nearest tenth
Answer:
4 number x value is 10 in
5 number x value is 12mi
use formula of h2=p2+b2
Let A(1, 1, 2 ), B ( 2, 3, 4 )and C ( 4, 2, 2 ) be three points in three dimensional vector.
find the coordinates of the point P such that it is on the line passing through A and B, and CP, AB are orthogonal. (Hint: Let the coordinates of P be ( x, y ,z ) Note that AP and AB are parallel.)
The coordinate of P is ( 14/5, 23/5, 28/5)
For the line AB, the direction vectors are
l= 2-1= 1, m=3-1= 2 and n= 4-2= 2
so, the equation of the line AB can be given by
[tex] \text{$\frac{x-1}{1}$ = $\frac{y-1}{2}$ = $\frac{z-2}{2}$ } [/tex]
let a point be P (x,y,z). lying on the line AB.
coordinate of the general point in AB can be given by
[tex] \text{$\frac{x-1}{1}$ = $\frac{y-1}{2}$ = $\frac{z-2}{2}$ = t } [/tex]
so, x= t+1, y= 2t+1, z= 2t+2
so, P (x,y,z) can be P (t+1, 2t+1, 2t+2). Another point is C ( 4, 2, 2 ).
the direction ratios of the line CP is
l1= t+1-4 = t-3
m1= 2t+1-2 = 2t-1
n1 = 2t+2-2 = 2t
Dot product of the direction ratios of two orthogonal lines are 0.
so, 1(t-3)+2(2t-1)+2(2t)=0
or, t=9/5
so, x= (9/5)+1 = 14/5
y= (9/5)*2 + 1= 23/5
z= (9/5)*2 + 2 = 28/5
so, the coordinate of P is ( 14/5, 23/5, 28/5).
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Instead of regular six-sided dice, some games use dodecahedronal - or 12-sided - dice. If you rolled a pair of dodecahedronal dice (each label 1 through 12) 100 times, about how many times would you expect the values on the two dice to add up to 4?
a) 1
b) 2
c) 3
d) 4
e) 6
f) 8
Instead of regular six-sided dice, some games use dodecahedronal - or 12-sided - dice. If you rolled a pair of dodecahedronal dice (each label 1 through 12) 100 times, you would expect the values on the two dice to add up to 4about 3 times. The correct answer is c) 3.
When rolling two dodecahedronal dice, there are a total of 12 x 12 = 144 possible outcomes. To find the probability of rolling a sum of 4, we need to look at the possible combinations that can result in a sum of 4:
1 + 3
2 + 2
3 + 1
There are a total of 3 possible combinations that can result in a sum of 4. Therefore, the probability of rolling a sum of 4 is 3/144 = 1/48.
If we roll the dice 100 times, we would expect to get a sum of 4 about (1/48) x 100 = 2.08333 times. Since we can't roll a fraction of a time, we can round this to the nearest whole number, which is 3. So, we would expect to roll a sum of 4 about 3 times out of 100 rolls.The correct answer is c) 3.
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Chase lives 1 1/5 blocks west of Ellie. They draw a number line to medel this situation. Because Ellie lives 1 1/2 blocks east of school, they give her house the coordinate 1 1/2. Chase incorrectly claims that the coordinate of his house is 2 7/10. What is the correct coordinate of Chase's house? What is Chase's likely error?
Chase's likely error is that he added instead of subtracting when finding his location on the number line.
What is the fractions?
A fraction is a way of representing a part of a whole or a ratio between two numbers. It consists of a numerator (top number) and a denominator (bottom number), separated by a line.
If Ellie's house is at 1 1/2 on the number line, and Chase lives 1 1/5 blocks west of Ellie, then the coordinate of Chase's house would be:
1 1/2 - 1 1/5 = 3/2 - 6/5
To subtract these fractions, we need a common denominator, which is 10:
(3/2) * (5/5) - (6/5) * (2/2) = 15/10 - 12/10 = 3/10
So Chase's house is located 3/10 blocks west of Ellie's house on the number line. Therefore, the correct coordinate for Chase's house is:
1 1/2 - 3/10 = 1 2/5
Chase's likely error is that he added instead of subtracting when finding his location on the number line.
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water sprinkler sends water out in a circular pattern. What is the area formed by the water pattern if it can spray out 24 feet in any direction? Use 3.14 for π.
1808.64 is the area formed by the water pattern if it can spray out 24 feet in any direction.
What is an area of a circle?The circle is a rounded shape without any edges or line segments. It has the geometric shape of a closed curve. The distance between the circle's points and its center is fixed. r2 is the area that a circle with radius r encloses. Here, the Greek letter stands for the constant proportion of a circle's circumference to its diameter,
Here, we have
Given: water sprinkler sends water out in a circular pattern. It can spray out 24 feet in any direction.
Since the water is being sent out in a circular pattern, the maximum distance at which the water is reaching out will be equal to the radius of the circle as the sprinkler is at the center of the circle.
The area of a circle is given as:
Area = πr²
Radius = 24 feet
Area = 3.14(24)²
Area = 1808.64
Hence, 1808.64 is the area formed by the water pattern if it can spray out 24 feet in any direction.
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Answer:
1808.64 is the area formed by the water pattern if it can spray out 24 feet in any direction.
step by step explanationWhat is an area of a circle?
The circle is a rounded shape without any edges or line segments. It has the geometric shape of a closed curve. The distance between the circle's points and its center is fixed. r2 is the area that a circle with radius r encloses. Here, the Greek letter stands for the constant proportion of a circle's circumference to its diameter,
Here, we have
Given: water sprinkler sends water out in a circular pattern. It can spray out 24 feet in any direction.
Since the water is being sent out in a circular pattern, the maximum distance at which the water is reaching out will be equal to the radius of the circle as the sprinkler is at the center of the circle.
The area of a circle is given as:
Area = πr²
Radius = 24 feet
Area = 3.14(24)²
Area = 1808.64
Hence, 1808.64 is the area formed by the water pattern if it can spray out 24 feet in any direction.
Solve: x-82-3
Graph the solutions.
The required graph represents that the solution to the given inequality x - 8 ≥ -3 is x ≥ 5.
What is linear inequality?A linear inequality is defined as mathematical statements that have a minimum of two terms containing variables or numbers that are not equal.
To solve for x:
x - 8 ≥ -3
x ≥ -3 + 8
x ≥ 5
So, the solution is x ≥ 5.
For the graph of the solution, draw a number line and mark a closed circle at 5 since the inequality includes 5. Then, shade to the right of 5 to represent all values of x that satisfy the inequality.
Thus, the required graph represents that the solution to the given inequality is x ≥ 5.
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three times a number, added to 4, is 40
Answer:
12
Step-by-step explanation:
12 × 3= 36
36 + 4= 40
so the answer is 12
Answer:
Not true with all numbers!
Step-by-step explanation:
see.Ex.3x3=9+4=13 not 40
Rewrite the quadratic function in standard form. \[ f(x)=\frac{f(x)=x^{2}-16 x+71}{} \] Give the vertex.
Rewriting the quadratic function [tex]\[ f(x)=(x-8)^{2}+7 \][/tex], we obtain as a result, vertex: [tex](8,7)[/tex].
How do we rewrite the quadratic function?To rewrite the quadratic function in standard form, we need to complete the square. The standard form of a quadratic function is [tex]\[ f(x)=a(x-h)^{2}+k \][/tex] where [tex](h,k)[/tex] is the vertex of the parabola.
Step 1: Separate the constant term from the rest of the equation.
[tex]\[ f(x)=x^{2}-16 x+71 \\\[ f(x)=(x^{2}-16 x)+71 \][/tex]
Step 2: Complete the square by adding and subtracting the square of half the coefficient of the x term inside the parenthesis.
[tex]\[ f(x)=(x^{2}-16 x+64)+71-64 \\\[ f(x)=(x-8)^{2}+7 \][/tex]
Now the equation is in standard form. The vertex of the parabola is (8,7).
Answer: [tex]\[ f(x)=(x-8)^{2}+7 \][/tex], vertex: [tex](8,7)[/tex].
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Rey predicted that the number of apple trees, x, planted in a farm could yield y=-20x^(2)+2800x pel year. How mary trees should be planted to produce the maximum number of apples per yer?
Mmaximum number of apples produced per year is 98000 when 70 apple trees are planted in the farm.
To find the maximum number of apples per year, we need to find the vertex of the quadratic equation y=-20x^(2)+2800x. The x-coordinate of the vertex can be found using the formula x=-b/(2a), where a=-20 and b=2800.
Plugging in the values for a and b, we get:
x=-2800/(2*-20)
x=-2800/(-40)
x=70
So, the maximum number of apples per year will be produced when 70 apple trees are planted in the farm. We can find the y-coordinate of the vertex by plugging x=70 back into the equation:
y=-20(70)^(2)+2800(70)
y=-20(4900)+196000
y=-98000+196000
y=98000
Therefore, the maximum number of apples produced per year is 98000 when 70 apple trees are planted in the farm.
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A person places $67700 in an investment account earning an annual rate of 7.7%, compounded continuously. Using the formula
V
=
P
e
r
t
V=Pe
rt
, where V is the value of the account in t years, P is the principal initially invested, e is the base of a natural logarithm, and r is the rate of interest, determine the amount of money, to the nearest cent, in the account after 7 years.
As a result, the balance in the account at the end of seven years is, to the closest cent, $116177.20.
What is amount ?The term "amount" in mathematics typically denotes a sum, total, or quantity. It could be a numerical value or some other kind of measurement. Depending on the context, the word "amount" may also be used to refer to quantity, total, sum, volume, or magnitude. \
The term "amount" is used in algebra and calculus to describe the outcome of an operation, such as the amount of change in a function or the quantity of a variable needed to satisfy an equation. In many disciplines, including finance, science, and engineering, the word "amount" is frequently used to refer to quantities or measurements.
given
The amount of money in the account after 7 years can be calculated using the formula V = Pe(rt), where V is the value of a account in t years, P is the principal implementation, e is the base of a natural log, and r is the rate of interest:
V = P*e(rt) = 67700*e(0.077*7)
[tex]V = 67700 * e^{(0.539) (0.539)[/tex]
V = 67700 * 1.716
V = 116177.20
As a result, the balance in the account at the end of seven years is, to the closest cent, $116177.20.
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If x=12 for one day of sales, use your equation to find the total number of pencils the
supply store sells.
Answer:
Step-by-step explanation:
If x represents one day of sales and x=12, then we can substitute that value into the equation to find the total number of pencils sold:
y = 160x
y = 160(12)
y = 1920
So the supply store sold a total of 1920 pencils on the day when x=12.
If a study found the time spent playing video games and ACT scores had a correlation of r= -1.91, you could reasonably conclude. (1 point)
A.
Clearly, r needs to be recalculated.
B.
Spending more time playing video games is associated with lower ACT scores.
C.
There is very little relationship between time spent playing video games played and ACT scores.
D. 191% of the variation in ACT scores can be explained by a linear relationship with the variation in time spent playing video games.
E.
There is a strong positive association between time spent playing video games and ACT scores.
Option A is correct, Clearly, r needs to be recalculated.
What is Statistics?Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data.
The correlation coefficient, r, measures the strength and direction of the linear relationship between two variables.
The value of r always falls between -1 and 1, where a negative value indicates a negative correlation (i.e., as one variable increases, the other tends to decrease), and a positive value indicates a positive correlation (i.e., as one variable increases, the other tends to increase).
However, the value of r cannot be less than -1 or greater than 1.
Therefore, a correlation coefficient of -1.91 is not a valid value.
Hence, Option A is correct, Clearly, r needs to be recalculated.
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Find the area of the shaded segment of the circle.
The area of the shaded segment of the circle is 5.79 square metre.
Area of SegmentThe area of a segment is equal to the area of a sector less the triangle-shaped portion.
A=[tex]\frac{1}{2}[/tex](∅-Sin∅)×[tex]r^{2}[/tex]
Circle definitionEvery point in the plane of a circle, which is a closed, two-dimensional object, is evenly separated from the centre. All lines that cross the circle join together to produce the line of reflection symmetry. Moreover, every angle has rotational symmetry around the centre.
Given;r=8m
∅=60°
Applying the formula's values, we obtain
A=[tex]\frac{1}{2}[/tex](∅-Sin∅)×[tex]r^{2}[/tex]
A=[tex]\frac{1}{2}[/tex]([tex]\frac{π}{3}[/tex]-Sin60°)×[tex]8^{2}[/tex]
A=5.79
Hence, the area of the shaded segment of the circle is 5.79 square metre.
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Let f(x)=20/1+9e-3x
What is the point of maximum growth rate for the logistic function f(x)? Round your answer to the nearest hundredth
answers:
(0,2)
(0.73,20)
(5.54,9)
(0.73,10)
The point of maximum growth rate is approximately (1.76, f'(1.76)) or (5.54, f'(5.54)) so the answer is (C) (5.54, 9).
What is the rate?
In mathematics, the rate is a measure of the change in one quantity with respect to another quantity. It is typically expressed as a ratio between the two quantities.
To find the point of maximum growth rate, we need to find the maximum value of the derivative of the function f(x).
First, we need to find the derivative of f(x):
[tex]f'(x) = (20 * 27e^{(-3x)}) / (1 + 9e^{(-3x)})^2[/tex]
To find the maximum value of f'(x), we set f''(x) = 0, where f''(x) is the second derivative of f(x):
[tex]f''(x) = (20 * 81e^{(-6x)} * (81e^{(-6x)} - 18e^{(-3x)} + 1)) / (1 + 9e^{(-3x)})^3[/tex]
Solving f''(x) = 0, we get:
[tex]81e^{(-6x)} - 18e^{(-3x)} + 1 = 0[/tex]
Letting [tex]y = e^{(-3x)}[/tex], we can rewrite the equation as:
81y² - 18y + 1 = 0
Using the quadratic formula, we get:
y = (18 ± √(18² - 4 * 81)) / (2 * 81) = 0.069 or 0.012
So, [tex]y = e^{(-3x)}[/tex] = 0.069 or 0.012
Solving for x, we get:
x = ln(0.069) / (-3) ≈ 1.76 or x = ln(0.012) / (-3) ≈ 5.54
Therefore, the point of maximum growth rate is approximately (1.76, f'(1.76)) or (5.54, f'(5.54)).
Now we need to calculate f'(1.76) and f'(5.54) to find the answer.
[tex]f'(1.76) = (20 * 27e^{(-5.28)}) / (1 + 9e^{(-5.28)})^2 = 9.62[/tex]
[tex]f'(5.54) = (20 * 27e^{(-16.62)}) / (1 + 9e^{(-16.62)})^2 = 1.01[/tex]
Rounding these values to the nearest hundredth, we get:
(1.76, 9.62) and (5.54, 1.01)
Therefore, the point of maximum growth rate is approximately (1.76, f'(1.76)) or (5.54, f'(5.54)) so the answer is (C) (5.54, 9).
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(Answer quick), Someone help me with this please?
Answer:
The angles are adjacent and the value of x is 63
Step-by-step explanation:
Find the inverse of each function:
y = log5 (4x + 1)
y = log3 (2^x + 8)
y = log4 (-2x + 9)
y = 5^x + 9/2
y = 10^x + 3/ -3
y = log5 (-4x + 10)
Taking the given functions, we will obtain the following inverse functions
1)[tex]f^-1(x) = (5^x - 1)/4[/tex]2) [tex]f^-1(x) = log2 (3^x - 8)[/tex]3)[tex]f^-1(x) = (9 - 4^x)/2[/tex]4)[tex]f^-1(x) = log5 (x - 9/2)[/tex]5) [tex]f^-1(x) = log10 (x - 3/ -3)[/tex]6) [tex]f^-1(x) = (10 - 5^x)/4[/tex]To find the inverse of each function, we need to switch the x and y values and solve for y. This will give us the inverse function.
1) [tex]y = log5 (4x + 1)[/tex]
Switch x and y:
[tex]x = log5 (4y + 1)[/tex]
Solve for y:
[tex]5^x = 4y + 1\\4y = 5^x - 1\\y = (5^x - 1)/4[/tex]
Inverse function:
2) [tex]y = log3 (2^x + 8)[/tex]
Switch x and y:
[tex]x = log3 (2^y + 8)[/tex]
Solve for y:
[tex]3^x = 2^y + 8\\2^y = 3^x - 8\\y = log2 (3^x - 8)[/tex]
Inverse function: [tex]f^-1(x) = log2 (3^x - 8)[/tex]
3) [tex]y = log4 (-2x + 9)[/tex]
Switch x and y:
[tex]x = log4 (-2y + 9)[/tex]
Solve for y:
[tex]4^x = -2y + 9\\-2y = 4^x - 9\\y = (9 - 4^x)/2[/tex]
Inverse function: [tex]f^-1(x) = (9 - 4^x)/2[/tex]
4) [tex]y = 5^x + 9/2[/tex]
Switch x and y:
[tex]x = 5^y + 9/2[/tex]
Solve for y:
[tex]5^y = x - 9/2\\y = log5 (x - 9/2)[/tex]
Inverse function: [tex]f^-1(x) = log5 (x - 9/2)[/tex]
5) [tex]y = 10^x + 3/ -3[/tex]
Switch x and y:
[tex]x = 10^y + 3/ -3[/tex]
Solve for y:
[tex]10^y = x - 3/ -3\\y = log10 (x - 3/ -3)[/tex]
Inverse function: [tex]f^-1(x) = log10 (x - 3/ -3)[/tex]
6) [tex]y = log5 (-4x + 10)[/tex]
Switch x and y:
[tex]x = log5 (-4y + 10)[/tex]
Solve for y:
[tex]5^x = -4y + 10-4y = 5^x - 10y = (10 - 5^x)/4[/tex]
Inverse function: [tex]f^-1(x) = (10 - 5^x)/4[/tex]
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Use elementary row operations to transform each augmented coefficient matrix to echelon form, then solve the system by back substitution. [5 Marks] 4x1 - 2x2 – 3x3 + x4 = 3 2x1 - 2x2 - 5x3 = -10 4x1 + x2 + 2x3 + x4 = 17 3x1 + x3 + x4 = 12
x1 = 0.75, x2 = 5, x3 = 0, x4 = 12
To transform the augmented coefficient matrix to echelon form using elementary row operations:
1. Subtract twice the first row from the second row.
2. Subtract the first row from the third row.
3. Subtract the first row from the fourth row.
4. Subtract the third row from the fourth row.
The resulting augmented coefficient matrix in echelon form is:
4x1 - 2x2 - 3x3 + x4 = 3
0x1 - 4x2 - 8x3 = -16
0x1 + x2 + 2x3 + x4 = 17
0x1 + 0x2 + x3 + x4 = 12
To solve the system by back substitution:
1. x4 = 12 (from the last equation)
2. x3 = 12 - x4 = 0 (substitute x4 from step 1 into the third equation)
3. x2 = (17 - x4) / (1) = 5 (substitute x4 from step 1 into the second equation)
4. x1 = (3 - 2*5 - 3*0) / (4) = 0.75 (substitute x2 and x3 from steps 2 and 3 into the first equation)
The solution is x1 = 0.75, x2 = 5, x3 = 0, x4 = 12.
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Consider the following trig function ????(????)=−34cos(8x−????/4)+150.
What is the maximum and minimum values of this function? Find an
angle in [0,????/4)
Maximum [ ] Angles that this occurs [
Maximum [184] Angles that this occurs [π/32 + nπ/4]
Minimum [116] Angles that this occurs [π/32 + (2n+1)π/8]
The maximum and minimum values of a trig function can be found by using the amplitude and the vertical shift of the function. The amplitude of the function is the absolute value of the coefficient of the cosine function, which is |-34| = 34. The vertical shift of the function is the constant term, which is 150. Therefore, the maximum value of the function is 34 + 150 = 184 and the minimum value of the function is 150 - 34 = 116.
To find an angle in the interval [0, π/4) where the maximum value occurs, we can use the fact that the cosine function has a maximum value of 1 when the angle is 0. Therefore, we can set the argument of the cosine function, 8x - π/4, equal to 0 and solve for x:
8x - π/4 = 0
8x = π/4
x = π/32
Since π/32 is in the interval [0, π/4), this is one of the angles where the maximum value occurs.
Therefore, the maximum value of the function is 184 and it occurs at angles x = π/32 + nπ/4, where n is an integer. The minimum value of the function is 116 and it occurs at angles x = π/32 + (2n+1)π/8, where n is an integer.
Maximum [184] Angles that this occurs [π/32 + nπ/4]
Minimum [116] Angles that this occurs [π/32 + (2n+1)π/8]
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